Interesting article on teaching math
toto
:Research points to a better way to teach math
:
:It's a typical math horror story: Your sixth-grade daughter
:gets confused in math class, but won't ask the teacher for help.
:She puts little effort into her homework and doesn't like to try
:new ways of learning math--all the while her math grades are
:dropping.
:
:What leads students to such academic self-destructive behavior?
:It may be the learning environment math teachers foster in their
:classrooms, suggests a new study.
:
:In the March issue of the Journal of Educational Psychology
:(Vol. 94, No. 1), University of Notre Dame researcher Julianne C.
:Turner, PhD, and her colleagues report that students are more
:likely to withdraw effort, resist novel learning approaches and
:avoid seeking help when they have math teachers who place
:low, rather than high importance on learning and mastery of
:math skills.
:
:The difference? Teachers who run low-mastery classrooms don't
:stop to clarify concepts or focus on whether students get the
:answers right, while those who run high-mastery classrooms
:emphasize the learning process by stressing that being unsure,
:learning from mistakes and asking questions are natural parts of
:learning.
:
:In the study, researchers surveyed 1,092 students on their
:avoidance strategies and then observed and audiotaped their
:math teachers in action. They found that the dialogue between
:teachers and students in the low- and high-mastery classrooms
:was quite different. For example, a high-mastery teacher asked
:her students if they remembered a previous concept, and added:
:"Don't look around and say to yourself, 'I'm not going to raise my
:hand because I don't want Jennifer to think I'm dumb.' If you don't
:remember, please raise your hand." When "Jason" did raise his
:hand, the teacher asked his classmate to explain the concept in
:"kid talk," then called Jason to the board to do a problem, offering
:cues to help him along the way.
:
:In contrast, a low-mastery teacher in the study seemed annoyed
:when students gave wrong answers, by, for instance, telling a
:student having trouble with a problem on the blackboard:
:"Serena, sit down. Barry, come up and fix that."
:
:The researchers found that students in the high-mastery classrooms
:were significantly less likely to use avoidance strategies than those
:in low-mastery classrooms.
:
:The authors also found that teachers who use appropriate doses of
:laughter and other motivational support were more likely to create
:environments where students felt comfortable asking for help.
:
:--D. SMITH
:
Dorothy
There is no sound, no cry in all the world
that can be heard unless someone listens ..
source unknown
Susan Umpleby
news:8stlhug6up272552i...@4ax.com...
> toto <scar...@wicked.witch> said:
<snop>
> I again question, do those people know what a high-mastery math class
> is ?
-------From reading the article, I believe the terms "high mastery" and "low
mastery" had nothing to do with the *level* of math being taught, but rather
referred to the numbers of kids in the classes that mastered the concepts
being taught. And these were 6th grade classes, not high school or college.
>
<snip>
> >:The authors also found that teachers who use appropriate doses of
> >:laughter and other motivational support were more likely to create
> >:environments where students felt comfortable asking for help.
>
> And those environments are hardly conducive to real learning,
> specially at a high-mastery level. Laughter is very distracting, it
> takes away the minimum required concentration. If a kid cannot survive
> a 50 minute classroom without having to be prodded into laughing on a
> constant basis, that kid doesn't belong in a high-mastery math class !
> Learning by laughter may be feasible in humanities, not in math.
>
> Sorry, Dorothy, no cigar.
>
------------And yet, in Japanese grade school classrooms, which are
typically very boisterous & much larger than ours, the kids consistently
outscore US kids in math (& other subjects as well). But then they foster a
learning atmosphere in which kids are not afraid to ask questions.
Jim Wayne
> "Susan Umpleby" <sump...@earthlink.net> said:
>
> >"Alberto Moreira" <junk...@moreira.mv.com> wrote in message
> >news:8stlhug6up272552i...@4ax.com...
> >> toto <scar...@wicked.witch> said:
> ><snop>
> >> I again question, do those people know what a high-mastery math class
> >> is ?
> >
> >-------From reading the article, I believe the terms "high mastery" and
"low
> >mastery" had nothing to do with the *level* of math being taught, but
rather
> >referred to the numbers of kids in the classes that mastered the concepts
> >being taught. And these were 6th grade classes, not high school or
college.
>
> DEPTH, not level, with INTENSITY, not level, with COMPULSIVITY, not
> level, with DETAIL, not level. Learning math is not about mastering
> concepts, there's a whole lot more behind it and before it. And even
> if one concentrates in learning concepts, they can be learned
>
>
SNIP
The paragraph above is incoherent.
What is "behind" mastering concepts? Mastering facts? I thought the poster
was maintaining that mastering facts was not enough. Of course concepts can
be learned. And if they are learned, are they not mastered? And if you
master things, whether concepts or facts, do you not achieve mastery? If
the class puts a high emphasis on achieving mastery, is it not a "high
mastery" class? And if it is a math class, does one not achieve high
mastery by learning math? Is the poster saying that one should not learn
math in math class? If not, what is he saying?
Jim Wayne
Susan Umpleby
> "Susan Umpleby" <sump...@earthlink.net> said:
>
> >"Alberto Moreira" <junk...@moreira.mv.com> wrote in message
> >news:8stlhug6up272552i...@4ax.com...
> >> toto <scar...@wicked.witch> said:
> ><snop>
> >> I again question, do those people know what a high-mastery math class
> >> is ?
> >
> >-------From reading the article, I believe the terms "high mastery" and
"low
> >mastery" had nothing to do with the *level* of math being taught, but
rather
> >referred to the numbers of kids in the classes that mastered the concepts
> >being taught. And these were 6th grade classes, not high school or
college.
>
> DEPTH, not level, with INTENSITY, not level, with COMPULSIVITY, not
> level, with DETAIL, not level. Learning math is not about mastering
> concepts, there's a whole lot more behind it and before it. And even
> different depths, with different intensity and compulsivity, and at
> different levels of detail.
--------???? The article used "high mastery" and "low mastery" to refer to
the numbers of kids in different classes who mastered the concepts being
taught them, whereas your statements were obviously referring to the level
or type of math, contrary to what you state above. And at its base, math
*is* about mastering concepts, otherwise no advancement is possible even in
the most basic aspects of math.
> >------------And yet, in Japanese grade school classrooms, which are
> >typically very boisterous & much larger than ours, the kids consistently
> >outscore US kids in math (& other subjects as well). But then they
foster a
> >learning atmosphere in which kids are not afraid to ask questions.
>
> Japanese teaching seems to be regimented and strict, and that is the
> secret.
----------While Japanese *high schools* are much more strict & what we might
call "regimented," that is certainly not true of their elementary and junior
high schools. On the contrary, classes there tend to be noisy and active.
Teachers encourage independent thought and classes are much more
student-driven with teachers there spending much less time lecturing, and
students learn to advance many alternate ways to solve problems.
I have had plenty of Japanese students in my classrooms, and
> they're far from being the type who are not afraid to ask questions.
> Actually, I haven't had one single Japanese student who would do
> anything but shut up, coil into him or herself, take copious notes,
> and sink into the work as if there was nothing or nobody else around.
> I have had Japanese professionals reporting to me, and their attitude
> was precisely the same.
----------which seems odd, when you consider that most Japanese large
businesses encourage their employees to take a significant role in
contributing to the running and management of their company.
Contrast it to your average boisterous and
> self-assured American kid, who is typically not afraid of coming out
> in a classroom even when it is very clear that he or she should not be
> doing so on account of manifest incompetence !
----------and yet, American kids have a real fear of giving the wrong answer
in class, of appearing "stupid" to their classmates. Or at the opposite end
of appearing too much of a "brain."
>
> It's precisely the other way around, people who tend to ask lots of
> questions are usually pretty clueless. The ones who master the subject
> don't ask that many questions, but when they do, they go to the core
> of the issue and they make the teacher quite uncomfortable, because
> they ask questions out of strong knowledge and mastery, questions that
> are usually hard to answer and denote insight that's plainly not
> available to those who don't have the mastery.
----------that may be your experience, however I found that very few teacher
had a problem with my asking pertinent questions--the one big exception
being a highly insecure *college* instructor of anthropology. And I had no
problem mastering most subjects at any level of schooling. To me, someone
who thinks they know it all or who never asks questions is usually the
"clueless" one--or at least lacking in interest and curiousity.
Jim Wayne
he is incorrect. Apparently he believes that unless a poster uses words
exactly as he wishes them to be used, their points have no validity. He
reminds me of Humpty Dumpty in _Through the Looking-Glass_.
For instance, the normal meaning of "concept" is exactly the kind of
abstraction he seems to favor, as several posters have pointed out. The
dictionary says it is a "general notion or idea," "an idea of something
formed by mentally combining all its characteristics or particulars," "a
directly conceived or intuited object of thought," or "a theme or image,
esp. as embodied in the design or execution of something." That is to say,
it is an abstraction, and the highest kind of abstraction. The poster seems
to be looking for a term for an even higher level of abstraction than
concept, and in normal English, there is no such word. One might direct him
to the theology of the Neo-Platonist--perhaps he is speaking of the
Demiurge.
It is wearying to read such repetitious and unhelpful niggling.
"That's glory for you,"
Jim Wayne
"Alberto Moreira" <junk...@moreira.mv.com> wrote in message
news:tscrhug3jknehclnk...@4ax.com...
> "Jim Wayne" <jhw...@bigfoot.com> said:
>
>
> >What is "behind" mastering concepts? Mastering facts? I thought the
poster
> >was maintaining that mastering facts was not enough. Of course concepts
can
> >be learned. And if they are learned, are they not mastered?
>
> It isn't about facts, and it isn't about concept. It's about PROCESS.
> And it's about ABSTRACTION. The concept is such a small thing that we
> sail over it as a matter of fact; and because math is about BUILDING
> the stuff, what's actually in it is uninteresting, so much for your
> facts.
>
> Read my post to Susan Umpleby, you will have a better perspective of
> how I see it.
>
> > And if you
> >master things, whether concepts or facts, do you not achieve mastery?
>
> In mathematics, neither fact nor concept will give you mastery. You
> need a very different set of capabilities.
>
> >If
> >the class puts a high emphasis on achieving mastery, is it not a "high
> >mastery" class?
>
> Given that the word "mastery" is being used in a sense I dispute, no,
> that's not it. Again, read my other post.
>
> >And if it is a math class, does one not achieve high
> >mastery by learning math?
>
> IF it is a math class. Emphasis on the MATH. If it is merely a home ec
> class disguised in math, no people don't learn math. They learn
> something else.
>
> >Is the poster saying that one should not learn
> >math in math class? If not, what is he saying?
>
> No. That's not what I heard the poster saying. And I've been
> interacting with Dorothy for years in this ng. I ask you to read
> carefully the claim, and my postings on it.
>
>
> Alberto.
>
>
Bob LeChevalier
>"Susan Umpleby" <sump...@earthlink.net> said:
>
>
>>--------???? The article used "high mastery" and "low mastery" to refer to
>>the numbers of kids in different classes who mastered the concepts being
>>taught them, whereas your statements were obviously referring to the level
>>or type of math, contrary to what you state above. And at its base, math
>>*is* about mastering concepts, otherwise no advancement is possible even in
>>the most basic aspects of math.
>
>you say is true, it confirms my point that they're not talking about
>mastery in mathematics as I understand the word.
They are talking about the concept of "mastery" as it the term is used in
education in general, and not merely in mathematics. It can apply to music
education as well. In mastery learning, you do not move on to the next topic
until you have demonstrated by performance that you have "mastered" the
current topic to a specified level (in music, I would guess this is the point
where the instructor says that the student has mastered the current piece or
exercise of focus well enough to start work on the next piece or exercise.
This does not mean that the student forgets the previous stuff or does not
continue to practice it, but there is some arbitrary level where one is ready
to go on.)
In this context, high-mastery means that the threshold of mastery of the
current topic is high. If one were using traditional grading to measure
mastery, this might mean requiring the student to get an A, or perhaps 90% or
better, in order to say he knows enough to proceed. In a "low mastery"
situation, the level could be much lower, 60% or 70%.
Traditional schooling in this country is not really "mastery" based, but to
the extent it is similar, it is low mastery. You are allowed to take the
next class if you pass the prior class (which means a D or better), or
sometimes get a C or better. Within a class, the teacher moves on to the
next topic even while some kids have not gotten the current one.
>My statements, again, did not refer to level but to approach and
>intensity. Mastery exists at every level.
"Mastery" in education parlance is not about intensity, and only indirectly
about approach.
>At its base, math is NOT about mastering concepts. The concept is a
>small part of it, the real issue is well beyond the concept level.
>Mastery in math is like mastery in a classical instrument: it's
>automatic, it's content free, it becomes part of our intuition, it's
>interwoven with our inner workings, it doesn't even bother about
>concept.
That actually is not unlike what they are saying "mastery" is. Supposedly you
have mastered a concept or topic when it is sufficiently understood that it
is so interwoven. But how do you measure mastery? "High mastery" requires a
higher level of demonstrated performance, and "low mastery" requires a lower
level of demonstrated performance.
>I call the first way the "naive" way - I call the second way the
>"mathematical" way. Note that in what I call the "mathematical" way,
>there's no real place for the naive concept of addition in a math
>class: it's irrelevant. Addition here is an abstract concept, DEFINED
>by the modeler, BUILT by the modeler. The only naive concept of
>addition we need is that addition is counting: 3+4 is, count to three,
>then count four more. Any more naive concept than that is a waste of
>time. You can teach that even to a first grader, what do you do with
>the rest of the K12 time ? Play games with rods ?
I think you would have to read the theory behind Cuisinaire rods before
people could discuss them with you. From what I have seen, they are a way to
develop mathematical thinking of the sort that you value with kids who are
not yet ready for pencil and paper approaches. The kids are learning to work
with models, without trying to correlate them with "reality", and the rods
serve as a physical manifestation of written symbols.
>In a quality class, at any level, students should read the book BEFORE
>the class. A classroom cannot be a place where the student gets the
>first contact with the material, or teaching becomes much harder, if
>not an outright gamble. Classrooms should be like labs, places where
>we reinforce knowledge, not places where we try to impart it - the
>place to acquire the initial knowledge is the book.
There is no K/12 classroom in America where the teacher can expect the kid to
have read the book before s/he teaches the class. At best the teacher can
"require" that the kids read the relevant part of the text as homework the
night before the test. But kids learn quickly that this is almost impossible
to verify, so such homework assignments are the first ones that they skip.
lojbab
Jim Wayne
> "Jim Wayne" <jhw...@bigfoot.com> said:
>
> >The poster seems to believe that I have not read his posts on this
thread;
> >he is incorrect. Apparently he believes that unless a poster uses words
> >exactly as he wishes them to be used, their points have no validity. He
> >reminds me of Humpty Dumpty in _Through the Looking-Glass_.
>
> professional in an area where I use them routinely. So, I'm using the
> words as they spring to me from my own professional field. If you
> don't like it, tough - if you can't talk basic math terms in a way
> that a professional of a math-intensive field can relate to, maybe you
> should re-evaluate your own meanings.
>
> >For instance, the normal meaning of "concept" is exactly the kind of
> >abstraction he seems to favor, as several posters have pointed out.
>
>
> Concept is NOT abstraction. In fact, in mathematics I can find hardly
> anything I can call "concept". There are DEFINITIONS - which are not
> concept. There are AXIOMS - which are not concept. There are RULES OF
> INFERENCE - which are not concept. And there is ABSTRACTION: the act
> of getting rid of irrelevancies. There's MODEL BUILDING: getting
> theorems by applying rules of inference to axioms. Which, again, is
> not concept.
>
> So, where the heck is that magic "concept" thing ? Why do I need to
> even bother about it ?
>
> >The
> >dictionary says it is a "general notion or idea," "an idea of something
> >formed by mentally combining all its characteristics or particulars," "a
> >directly conceived or intuited object of thought," or "a theme or image,
> >esp. as embodied in the design or execution of something."
>
> the MATHEMATICAL meaning of the word that carries some weight, and
> even then, only to the extent of the definitions we use in our model -
> and those definitions, mind you, change from model to model.
>
> So, for example, you could say that the "concept" of two lines being
> parallel is that they never meet. But that's no concept, that's either
> an axiom or the consequence of a chain of inference drawn on a set of
> axioms: and I can construct a geometry where lines can be parallel yet
> they will meet somewhere. It's merely a question of where I place my
> axioms, and how I set up my model.
>
> >That is to say,
> >it is an abstraction, and the highest kind of abstraction.
>
> we do not want to insert in the model we're building. Emphasis on
> BUILDING - what abstraction consists of changes from model to model,
> what's abstraction here may not be abstraction there, and vice-versa,
> yet both are equally valid pieces of mathematical thought.
>
> >The poster seems
> >to be looking for a term for an even higher level of abstraction than
> >concept, and in normal English, there is no such word. One might direct
him
> >to the theology of the Neo-Platonist--perhaps he is speaking of the
> >Demiurge.
>
> cornerstones of mathematical philosophy. If nothing else, normal
> English is NOT necessarily a valid vernacular for mathematical
> thought.
>
> >It is wearying to read such repetitious and unhelpful niggling.
>
> mathematics. Hence my disagreement. And you know what, until you
> manage to stand on my platform and understand the meaning of the terms
> as I'm using it, you may be right in feeling that this debate is not
> very useful.
>
I entirely agree with your final comment.
Jim Wayne
Bob LeChevalier
>Bob LeChevalier <loj...@lojban.org> said:
>>They are talking about the concept of "mastery" as it the term is used in
>>education in general, and not merely in mathematics. It can apply to music
>>education as well. In mastery learning, you do not move on to the next topic
>>until you have demonstrated by performance that you have "mastered" the
>>current topic to a specified level (in music, I would guess this is the point
>>where the instructor says that the student has mastered the current piece or
>>exercise of focus well enough to start work on the next piece or exercise.
>>This does not mean that the student forgets the previous stuff or does not
>>continue to practice it, but there is some arbitrary level where one is ready
>>to go on.)
>
>consequence of your accumulated technique: if you have enough
>technique, the piece comes, and you take for granted that the notes
>will be there, and you can forget technique at a conscient plane and
>concentrate on the interpretation. And here, again, there's little
>direct connection between thought and action, there seems to be a
>direct path from musical conception to one's fingers that doesn't go
>through one's rational thinker ! It's all about automated responses.
And when you can do it automatically, that is probably "high mastery". "Low
mastery" would be more like what my own kids do - they go on to something new
when they are satisfied that they've done "good enough" and thus seldom reach
automaticity.
>>In this context, high-mastery means that the threshold of mastery of the
>>current topic is high. If one were using traditional grading to measure
>>mastery, this might mean requiring the student to get an A, or perhaps 90% or
>>better, in order to say he knows enough to proceed. In a "low mastery"
>>situation, the level could be much lower, 60% or 70%.
>
>But measuring precisely what? My point is, if you're not measuring
>that "technique", you're probably not measuring mastery.
Correct. A high mastery math class requires people to master the technique
to a high degree of proficiency before moving on, whereas low mastery tends
to be satisfied with vaguer demonstrations of understanding.
>The concept,
>you see, is more or less irrelevant, but can you use the stuff ? Make
>it do things for you ? You can call anyone's level "high mastery" if
>only you measure something they're good at.
But a *class* is high mastery, if it expects the students to master
techniques to a higher performance level (mastery is always measured by
demonstrated performance).
>>I think you would have to read the theory behind Cuisinaire rods before
>>people could discuss them with you. From what I have seen, they are a way to
>>develop mathematical thinking of the sort that you value with kids who are
>>not yet ready for pencil and paper approaches. The kids are learning to work
>>with models, without trying to correlate them with "reality", and the rods
>>serve as a physical manifestation of written symbols.
>
>I believe rods, or any other prop, is something we should only use in
>exceptional cases, with abnormally weak students.
We're talking about kids who are still learning to read, which means that
they are mathematically speaking rather weak indeed.
>>There is no K/12 classroom in America where the teacher can expect the kid to
>>have read the book before s/he teaches the class. At best the teacher can
>>"require" that the kids read the relevant part of the text as homework the
>>night before the test. But kids learn quickly that this is almost impossible
>>to verify, so such homework assignments are the first ones that they skip.
>
>Well, too bad, because that makes effective teaching very difficult.
It does indeed, but the teachers who discuss the problems of teaching have to
bear in mind that they have no real way to enforce discipline on the
students, and usually nowhere near the support needed from parents to expect
kids to prepare at the level you would require. Indeed I suspect that many
teachers in the K/12 grades have to be satisfied if all the kid come to class
with a pencil and notepaper every day.
lojbab
Bob LeChevalier
>have been having great success with converting my classes into lab
>sessions. I sometimes wonder if I could do the same with math, but of
>course this would require such a radical rethinking of the whole thing
>that I can't summon enough energy and faith to float a first balloon.
>I believe I have said it before, I'm believing less and less in
>classroom teaching of any sort.
Indeed this is more or less what they've tried to do in the lower grades.
Cuisinaire rods are more or less a kind of laboratory approach to math, one
which seems a bit more rigorously designed than some other manipulative
techniques.
Efforts to go to what the education business calls "mastery learning" really
amounts to moving towards the approach you've described for music teaching,
except that the lessons are more like "master classes" than like individual
lessons, allowing them to work with the higher student/teacher ratios that
the schools have to deal with.
lojbab
Seveigny
eldest brother. Alberto's remarks puzzled me so I consulted my brother, who
has a Ph.D.. in Experimental Mathematics and an MS in Computer Science. He
teaches computer science at Stanislaus State.
On the subject of a dictionary definition of concept not measuring up to a
mathematical definition of concept, he said.
I find this in Merriam-Webster --
1 : something conceived in the mind : THOUGHT, NOTION
2 : an abstract or generic idea generalized from particular instances
This seems to fit well with the idea of abstraction being important in
mathematics.
On the subject of whether concepts exist in mathematics, he said
Certainly axioms, rules of inference, abstraction, and so on are quite
important in mathematics. What I don't understand is why he says they
are "not concept" -- what's his objection to that term? What do we
mean here by "concept?"
Finally, in response to one of Alberto's statements, he remarked
So, for example, you could say that the "concept" of
> two
> lines being parallel is that they never meet. But that's no concept,
> that's
> either an axiom or the consequence of a chain of inference drawn on a
> set of
> axioms: and I can construct a geometry where lines can be parallel yet
> they
> will meet somewhere. It's merely a question of where I place my axioms,
> and
> how I set up my model."
Well there are different geometry's -- in Euclidean geometry, parallel
lines don't meet. In other geometries, they may meet -- for example if
"space" is curved in the right way. Of course, you can't just choose
axioms at random -- else those rules of inference may lead to
contradiction.
I don't know if this helpful to anyone else, but it helped me out.
Cate
"Bob LeChevalier" <loj...@lojban.org> wrote in message
news:3umvhu43hsi5l5mno...@4ax.com...
sf
on such a high horse, I just stopped reading him. I figured: Hell, I
don't need to understand or agree with his point. I just teach
addition and subtraction.
I often wonder what he and Herman are like as teachers when I read
them. Do they turn kids on to higher math or away from it?
<G>
Barbara
``````````````````````````
On Mon, 01 Jul 2002 05:35:16 GMT, "Seveigny" <sarra...@attbi.com>
wrote:
toto
>Thanks. You brother sounds much more reasonable than Alberto. He was
>on such a high horse, I just stopped reading him. I figured: Hell, I
>don't need to understand or agree with his point. I just teach
>addition and subtraction.
>
>I often wonder what he and Herman are like as teachers when I read
>them. Do they turn kids on to higher math or away from it?
>
><G>
>Barbara
>``````````````````````````
Since neither of them teaches young children, it is hard
to tell what they would do or if they could teach it to the
average elementary school child (not the 10% who are
talented already)
Seveigny
"sf" <s...@pipeline.com> wrote in message
news:3d20992d...@news.pipeline.com...
> Thanks. You brother sounds much more reasonable than Alberto. He was
> on such a high horse, I just stopped reading him. I figured: Hell, I
> don't need to understand or agree with his point. I just teach
> addition and subtraction.
He's very reasonable. A couple of years ago, I asked the chairman of our
math department to explain what was taught in a class entitled Fibonnacci
(sp?) numbers. He refused to do so, saying I couldn't possibly understand
his explanation. I asked my brother and he explained it clearly and
concisely.
> I often wonder what he and Herman are like as teachers when I read
> them. Do they turn kids on to higher math or away from it?
Good question. Alberto's condescending tone ticked me off but his
explanations didn't make sense either.
Cate
Herman Rubin
Alberto Moreira <junk...@moreira.mv.com> wrote:
>toto <scar...@wicked.witch> said:
>>What do you think that discovery learning sessions in math at the
>>K-12 level are? They are guided labs, Alberto and they do work
>>better than lecture and drill.
>Dorothy, lab is not about discovery. Lab is about learning by doing,
>and about learning by building, within an exceedingly controlled
>environment where the student has nearly zero option to engage in
>anything except that precise item that the teacher tells them to.
> The difference between a lab session and a classroom is not the
>"discovery" aspect - I find it irrelevant - but the fact that in the
>lab the teacher can direct every finger movement of the student.
>In a classroom, I talk and they listen - and I don't know if I have
>their minds. In a lab, I have their bodies, their bodies have their
>minds, and hence I have their minds.
>Again, it has nothing to do with discovery.
In the labs I have taken, we were told what experiments to
perform, and even what results to find. There was neither
direction nor discovery.
>Alberto.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Herman Rubin
>>Indeed this is more or less what they've tried to do in the lower grades.
>>Cuisinaire rods are more or less a kind of laboratory approach to math, one
>>which seems a bit more rigorously designed than some other manipulative
>>techniques.
>The problem with teaching math this way is, lab is not about using
>props.
It depends on the lab. I believe that the Cuisinaire rods
are a reasonable means of getting accross the "distance"
concept of numbers, and could be used for the cardinal one
as well. In any case, it is far superior to memorizing the
addition tables.
................
>>Efforts to go to what the education business calls "mastery learning" really
>>amounts to moving towards the approach you've described for music teaching,
>>except that the lessons are more like "master classes" than like individual
>>lessons, allowing them to work with the higher student/teacher ratios that
>>the schools have to deal with.
>I don't believe that the education establishment has a clue about what
>goes on in your typical music class. If they did, they'd be speaking
>of very different things.
The educationists have no idea how unimportant drill is.
Mastery learning rarely is good, but only someone who
understands the subject can have a reasonable idea when
can safely go on and come back to a partially understood
point. It is rarely a matter of memorization or being
able to do the routine.
Herman Rubin
Alberto Moreira <junk...@moreira.mv.com> wrote:
>toto <scar...@wicked.witch> said:
>>Since neither of them teaches young children, it is hard
>>to tell what they would do or if they could teach it to the
>>average elementary school child (not the 10% who are
>>talented already)
Mathematicians, and even students who learned mathematical
concepts in sound abstract courses, were able to teach the
"new math" quite well to the average elementary school child.
Even if the tens of thousands of children on whom it was
tested were not quite representative, that should be
sufficient evidence that it can be done for at least a
large proportion.
>Children don't remain young forever, and the period where we can do
>thing nice and soft is two or three years at most. After that, reality
>must sink in: we have lots to teach and not enough time to teach it.
>So, if a teacher of young children doesn't teach with an eye on the
>future, he or she is not doing the kids a favor.
>Alberto.
This is a major problem with the "educational" system.
There is no eye on doing things efficiently; few teachers
realize that they do not know enough for the children.
Even fewer seem to be willing to let the children zoom
ahead.
We cannot have a decent educational system as long as
anyone is allowed to keep a child from learning, even if
not all can do it at that rate.
Herman Rubin
>Thanks. You brother sounds much more reasonable than Alberto. He was
>on such a high horse, I just stopped reading him. I figured: Hell, I
>don't need to understand or agree with his point. I just teach
>addition and subtraction.
>I often wonder what he and Herman are like as teachers when I read
>them. Do they turn kids on to higher math or away from it?
I do not think that Alberto and I are that much alike as
teachers, or in the way we understand mathematics. I
concentrate nor on teaching methods, but on concepts and
basic principles, and am personally appalled at the way
textbooks, etc., are written, with the rather simple
concepts thoroughly concealed by cute proofs, algorithms
without understanding where the understanding could be
easily given, lots of use of sometimes confusing special
cases, when the general case is often simpler.
Concepts are not verbal descriptions, and cannot be
understood in that manner. Nor can they be "led up to" by
special cases which are not general enough. There are at
least three concepts for the integers, and while the
objects with these properties are isomorphic, the concepts
are not the same, and each needs to be understood for what
it is.
I have published in many areas of mathematics, probability,
and statistics, and have contributed to the foundations.
In all of these, learning calculation does not at all help
in understanding. I seem to have a very rare ability (I
am complaining that not enough others have it) to be aware
that, while I can use something extremely well, this does
not mean that I understand it.
The only kids I have taught are my own, one of whom became
a mathematician, and the other does not hesitate to use
"higher mathematics".
><G>
>Barbara
>``````````````````````````
>On Mon, 01 Jul 2002 05:35:16 GMT, "Seveigny" <sarra...@attbi.com>
>wrote:
>>Since we've returned to this topic, I'd like to share the thoughts of my
>>eldest brother. Alberto's remarks puzzled me so I consulted my brother, who
>>has a Ph.D.. in Experimental Mathematics and an MS in Computer Science. He
>>teaches computer science at Stanislaus State.
>>On the subject of a dictionary definition of concept not measuring up to a
>>mathematical definition of concept, he said.
>>I find this in Merriam-Webster --
>>1 : something conceived in the mind : THOUGHT, NOTION
>>2 : an abstract or generic idea generalized from particular instances
Many mathematical concepts come from this second, but they
are not mere generalizations, but have "life of their own."
The concepts are typically easier to understand, and make
the special cases themselves easier to understand later.
The converse direction is slower and harder.
Nobody who has an understanding of the integers will have
difficulty with induction; it used to be that this was used
heavily in college algebra and calculus, but it cannot be
so used today. The limit concept should come in with the
presentation of infinite decimals, not just given lip service
in the calculus course, which should be based on it.
>>This seems to fit well with the idea of abstraction being important in
>>mathematics.
>>On the subject of whether concepts exist in mathematics, he said
>>Certainly axioms, rules of inference, abstraction, and so on are quite
>>important in mathematics. What I don't understand is why he says they
>>are "not concept" -- what's his objection to that term? What do we
>>mean here by "concept?"
These are mostly not concepts. What is a group is not merely
a characterization, but a concept, and it is the same concept
no matter which "definition" is given. There is a concept of
integral, and the present method of teaching it is quite poor;
as I have to remind graduate students, measure is discrete and
limits of discrete, not based on "area under the curve", which
is approximated by discrete.
toto
Rubin) wrote:
>In article <ks83iukp34oteagq4...@4ax.com>,
>Alberto Moreira <junk...@moreira.mv.com> wrote:
>>toto <scar...@wicked.witch> said:
>
>
>>>Since neither of them teaches young children, it is hard
>>>to tell what they would do or if they could teach it to the
>>>average elementary school child (not the 10% who are
>>>talented already)
>
>Mathematicians, and even students who learned mathematical
>concepts in sound abstract courses, were able to teach the
>"new math" quite well to the average elementary school child.
The students they taught were not average elementary school
students. This was done in University Lab schools which don't
have average kids.
>Even if the tens of thousands of children on whom it was
>tested were not quite representative, that should be
>sufficient evidence that it can be done for at least a
>large proportion.
>
Please site the 10s of thousands of kids taught by mathematicians
who were not teachers? The first kids *may* have been taught
by mathematicians, the next groups were taught by teachers who
were trained by those mathematicians, not by the mathematicians
themselves.
>>Children don't remain young forever, and the period where we can do
>>thing nice and soft is two or three years at most. After that, reality
>>must sink in: we have lots to teach and not enough time to teach it.
>>So, if a teacher of young children doesn't teach with an eye on the
>>future, he or she is not doing the kids a favor.
<snip>
Bob LeChevalier
>In article <l773iuolek1oc1ain...@4ax.com>,
>Alberto Moreira <junk...@moreira.mv.com> wrote:
>>Bob LeChevalier <loj...@lojban.org> said:
>>>Indeed this is more or less what they've tried to do in the lower grades.
>
>>>Cuisinaire rods are more or less a kind of laboratory approach to math, one
>>>which seems a bit more rigorously designed than some other manipulative
>>>techniques.
>
>>The problem with teaching math this way is, lab is not about using
>>props.
>
>It depends on the lab. I believe that the Cuisinaire rods
>are a reasonable means of getting accross the "distance"
>concept of numbers, and could be used for the cardinal one
>as well.
The basic experiments (which fit Alberto's description that a lab should have
specific instructions and not be mere exploration) also seem to support the
ordinal concept as well, wherein you sequence the rods in terms of length,
and observe that each successive rod differs from the prior one by a unit
length equal to the shortest rod. This can be generalized also by building
sequences where the length differs by 2 units, etc.
>>>Efforts to go to what the education business calls "mastery learning" really
>>>amounts to moving towards the approach you've described for music teaching,
>>>except that the lessons are more like "master classes" than like individual
>>>lessons, allowing them to work with the higher student/teacher ratios that
>>>the schools have to deal with.
>
>>I don't believe that the education establishment has a clue about what
>>goes on in your typical music class. If they did, they'd be speaking
>>of very different things.
I'm not a member of the education establishment, and I've come to what I
presume are "typical music classes" as a non-musician who has been sitting
there for a couple of years during my kids' lessons and observing what goes
on, partly in light of Alberto's comments over the years. I've also seen a
couple of master classes where groups of students show their stuff to each
other and a superior instrumentalist who is not their regular instructor.
Both are ways of dealing with students are individually at different places
in their studies, and mastery learning assumes that they are at different
places at their studies.
I don't know that all mastery learning classroom sessions would necessarily
be like musical master classes, but they COULD use such a model, and I recall
some instances, even back when I was a student, when we studied a topic in a
way that now seems like it was patterned after that model.
I also have to say that my experience watching maybe a dozen different music
instructors, is that there isn't much that is necessarily "typical" in music
classes. Those things that I have generalized from them, however, CAN be
emulated in a mastery learning curriculum, but probably not exactly.
The bottom line is that individual music lessons are too inefficient for
schooling, and that is evidenced by the economics of music lessons. At the
$20 for an hour lesson a week that is typical around here, a highly paid
area, a music teacher cannot approach a teacher's average salary unless he
can fill 40 working hour slots a week, and takes no vacations (and this
provides no benefits and the music teacher has to cover the higher
self-employment social security tax rate). And few teachers come close to
that volume (which I imagine would be incredibly draining).
>The educationists have no idea how unimportant drill is.
And yet, in music, drill is critical and the difference between one level and
the next seems to at what level you do your drilling. Transferring this to
math, a person who is not drilling arithmetic facts may not progress to
algebra, a person who drills on proving Euclidean theorems will be able to
move on to higher proofs.
>Mastery learning rarely is good, but only someone who
>understands the subject can have a reasonable idea when
>can safely go on and come back to a partially understood
>point.
Which is why mastery learning does not presume to allow you to move on when
you only have a partial understanding.
lojbab
toto
Rubin) wrote:
>The educationists have no idea how unimportant drill is.
>Mastery learning rarely is good, but only someone who
>understands the subject can have a reasonable idea when
>can safely go on and come back to a partially understood
>point. It is rarely a matter of memorization or being
>able to do the routine.
Is Alberto an educationist, then? It seems to me that his position
on practice would be an emphasis on drill here. With playing
an instrument or touch typing for that matter, you practice until
it is almost as if your body has the memory to perform the task.
Alberto has said some things that make me believe that he thinks
that mathematical knowledge must also be somatic in some way.
My son would say that it's mathematical insight and intuition and
it's not something that is directly taught. Alberto seems to have
some idea that there is a correspondence between musical practice
and mathematical intuition if my reading of his posts is not
mistaken.
toto
Rubin) wrote:
>In article <5v63iusm4b9f580gj...@4ax.com>,
>Alberto Moreira <junk...@moreira.mv.com> wrote:
>>toto <scar...@wicked.witch> said:
>
>
>>>What do you think that discovery learning sessions in math at the
>>>K-12 level are? They are guided labs, Alberto and they do work
>>>better than lecture and drill.
>
>>Dorothy, lab is not about discovery. Lab is about learning by doing,
>>and about learning by building, within an exceedingly controlled
>>environment where the student has nearly zero option to engage in
>>anything except that precise item that the teacher tells them to.
>
>> The difference between a lab session and a classroom is not the
>>"discovery" aspect - I find it irrelevant - but the fact that in the
>>lab the teacher can direct every finger movement of the student.
>
>>In a classroom, I talk and they listen - and I don't know if I have
>>their minds. In a lab, I have their bodies, their bodies have their
>>minds, and hence I have their minds.
>
>>Again, it has nothing to do with discovery.
>
>In the labs I have taken, we were told what experiments to
>perform, and even what results to find. There was neither
>direction nor discovery.
>
And the experiments lead to discovery by the students unless
you never had an *aha* moment when you did perform and
experiment and saw that the result was as expected. And, of
course, in high school physics the results of many labs don't
actually come out the way they are supposed to for various reasons,
so fudging becomes the norm and the students learn nothing from
that at all.
In elementary school, there is more leeway to do labs that encourage
that *aha* moment in the student's mind.
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On 2 Jul 2002 09:37:22 -0500, hru...@odds.stat.purdue.edu (Herman
>Rubin) wrote:
>>In article <ks83iukp34oteagq4...@4ax.com>,
>>Alberto Moreira <junk...@moreira.mv.com> wrote:
>>>toto <scar...@wicked.witch> said:
>>>>Since neither of them teaches young children, it is hard
>>>>to tell what they would do or if they could teach it to the
>>>>average elementary school child (not the 10% who are
>>>>talented already)
>>Mathematicians, and even students who learned mathematical
>>concepts in sound abstract courses, were able to teach the
>>"new math" quite well to the average elementary school child.
>The students they taught were not average elementary school
>students. This was done in University Lab schools which don't
>have average kids.
University Lab schools are not that great. They are not
selective on the basis of ability.
Much of it was done in private schools, but these were in
no sense academic private schools.
>>Even if the tens of thousands of children on whom it was
>>tested were not quite representative, that should be
>>sufficient evidence that it can be done for at least a
>>large proportion.
>Please site the 10s of thousands of kids taught by mathematicians
>who were not teachers? The first kids *may* have been taught
>by mathematicians, the next groups were taught by teachers who
>were trained by those mathematicians, not by the mathematicians
>themselves.
They were taught by teachers who were EDUCATED by
mathematicians. The subsequent results showed that
teachers who do not understand mathematics cannot be
TRAINED to teach mathematical concepts.
Donna Metler
--
Donna DeVore Metler
Orff/ Band/Choral music, Lester Focused Literacy School
Mother to Angel Brian Anthony, 01/01/02 (22 weeks, severe PE/HELLP syndrome)
"Bob LeChevalier" <loj...@lojban.org> wrote in message
news:e0v3iug2j1sk8c3vu...@4ax.com...
$20/hr is also extremely LOW for individual private lessons. The Academy
gets $500 per semester for one private student at the beginning/intermediate
level, which pays for 1 30 minute lesson a week, one 1 hr group class a
week, and one recital. As the private teacher, I get $400 of that, and the
group teacher gets $25 per student (but has about 20-30 students).
Full hour classes are $1000/semester, of which I get $850.
The only place I know which charges significantly less is the University
Community music school, where college students teach lessons as part of
their required experience for a music education or music
performance/pedagogy degree, under the supervision of the university
professor.
It is next to impossible to make a full-time job out of teaching privately,
because most of your students are in school during the day, and there are
only so many hours in the evening and on weekends. If you're not a strings
teacher or piano teacher, it is very difficult to get enough students, too,
because while there are hundreds of saxophonists (for example, since I am a
sax teacher) in a good sized city, only a very small percentage of students
take lessons privately, and once a student is serious enough to be planning
to go to college, it is usually much better for the student to enroll in
private lessons as a university/conservatory class, if at all possible, and
get college credit and, more important, lessons with someone who has
professional credentials which will mean more to universities or
conservatories. Serious students also usually are attending music camps at
places like Carnegie Mellon or Interlochen during the summer.
>
> >The educationists have no idea how unimportant drill is.
>
> And yet, in music, drill is critical and the difference between one level
and
> the next seems to at what level you do your drilling. Transferring this
to
> math, a person who is not drilling arithmetic facts may not progress to
> algebra, a person who drills on proving Euclidean theorems will be able to
> move on to higher proofs.
Just plain drill isn't enough, though. It is very focused drill, and is very
prescriptive. Part of learning to be a musician-and the difference between a
serious music student and a school band member, is that the serious student
learns how to self-teach. The actual lesson is designed to refine and
expand, not present new material.
See, in music once you get past the first few years, you're really not
learning new skills. You already know how to read music, how to finger most
of the instrument (although you'll be learning new tricks until the end of
your days-multiphonics and the altissimo register on the saxophone both are
things which the instrument wasn't really designed to do-but a good player
can learn and master anyway), and similar skills. What you don't have is the
facility in the mind and the fingers, and on a wind instrument or voice, in
the mouth, lungs, diaphragm and throat, to do all the higher level skills.
This is what can be learned through practice.
In math, though, you're constantly learning new skills, and what is learned
in one branch doesn't always carry over to others. A student who fails
algebra can still do quite well in Geometry, for example. Arithmetic isn't
terribly relevant to a theoretical mathematician.
>
> >Mastery learning rarely is good, but only someone who
> >understands the subject can have a reasonable idea when
> >can safely go on and come back to a partially understood
> >point.
>
> Which is why mastery learning does not presume to allow you to move on
when
> you only have a partial understanding.
>
And, in music, you overpractice-that is, you don't move on when you have a
piece mastered. You move on when you have a piece mastered to the point that
you don't even have to think anymore to play it. My instructor once told me
that if you were still sweating, you didn't know it yet. (This was about a
fiendishly difficult concerto).
You've got to remember that music, and music instruction, isn't for everyone
once you get past the most basic levels. The music programs which are
designed for all children, such as Orff, Kodaly, Dalcroze, and, to a degree,
Suzuki and Yamaha, still have some mastery components, but the amount of
exposure without mastery is much greater too. Even high school music tends
to be a combination of mastery and exposure. Only the serious students
really get into the "know it, live it, be it" mentality-and I think this is
one reason why there were so many excellent amateurs who weren't music
majors at the college level. Incidentally, almost all of the non-music
majors who were good enough that they could have been music majors were
either math, hard science, or engineering majors-lots of chemistry majors at
my school, for some reason.
> lojbab
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <l773iuolek1oc1ain...@4ax.com>,
>>Alberto Moreira <junk...@moreira.mv.com> wrote:
>>>Bob LeChevalier <loj...@lojban.org> said:
>>>>Indeed this is more or less what they've tried to do in the lower grades.
>>>>Cuisinaire rods are more or less a kind of laboratory approach to math, one
>>>>which seems a bit more rigorously designed than some other manipulative
>>>>techniques.
>>>The problem with teaching math this way is, lab is not about using
>>>props.
>>It depends on the lab. I believe that the Cuisinaire rods
>>are a reasonable means of getting accross the "distance"
>>concept of numbers, and could be used for the cardinal one
>>as well.
>The basic experiments (which fit Alberto's description that a lab should have
>specific instructions and not be mere exploration) also seem to support the
>ordinal concept as well, wherein you sequence the rods in terms of length,
>and observe that each successive rod differs from the prior one by a unit
>length equal to the shortest rod. This can be generalized also by building
>sequences where the length differs by 2 units, etc.
This is not ordinal, but cardinal. And it requires realizing
that units combine in the counting manner.
The ordinal approach requires that the comparison of one rod
must be made with an initial segment of the other. How likely
is it for a child to get this? I cannot see how to present a
solid foundational approach to the integers without using the
ordinal structure at some time, but I do not believe that this
was integrated into the "intuitive" approach of the new math.
The ordinal approach is counting by succession, not by arbitrary
assignment as in the cardinal approach.
Donna Metler
--
Donna DeVore Metler
Orff/ Band/Choral music, Lester Focused Literacy School
Mother to Angel Brian Anthony, 01/01/02 (22 weeks, severe PE/HELLP syndrome)
"toto" <scar...@wicked.witch> wrote in message
news:u714iuk420q0p1pp3...@4ax.com...
> On Tue, 02 Jul 2002 08:34:59 -0400, Alberto Moreira
> <junk...@moreira.mv.com> wrote:
>
> >I don't believe that the education establishment has a clue about what
> >goes on in your typical music class. If they did, they'd be speaking
> >of very different things.
>
> Since I have taken individual music lessons, but not the type of
> master classes that are being described by you, I wonder if you
> can bring it to us by describing a class from beginning to end
> specifically. Take a single class from beginning to end. What
> is each student doing, what is the teacher doing in detail for that
> hour?
>
> Dorothy
Here's how we do master classes, and how the ones I've taken go:
A group of students, at similar levels, usually 5-6, no more than 10 and a
clinician come into a fairly small room. Each student brings one piece of
music which he/she is working on, and plays it for the group. After it is
finished, the group and the clinician pick it to pieces-and it can be really
brutal. After this, the student tries the piece again, applying the
suggestions. You leave exhausted, stressed-but with LOTS of things to work
on. I've seen students, especially those who thought they were absolutely
wonderful, brought to tears in master classes before.
Usually you pick the piece based on the clinician's expertise. For example,
you wouldn't bring a highly technical piece to a session by Sigurd Rascher,
because he is a very lyrical player. You would bring a piece where you're
working on expressive playing. You also won't see a beginning student at a
master class at this level.
Master Classes are generally only done by conservatory level students or
professionals-so high school age and above. It is not odd to have master
classes where each participant him/herself could easily lead the group-and
participating in one of those is wonderful.
Jim Wayne
news:aft7la$1s...@odds.stat.purdue.edu...
Please cite the research that shows these results. [1]
Jim Wayne
[1] You have made the claim, it is your duty to support it or withdraw it.
Jim Wayne
"toto" <scar...@wicked.witch> wrote in message
> On Tue, 02 Jul 2002 09:00:30 -0400, Alberto Moreira
> >>to tell what they would do or if they could teach it to the
> >>average elementary school child (not the 10% who are
> >>talented already)
> >
> >thing nice and soft is two or three years at most. After that, reality
> >must sink in: we have lots to teach and not enough time to teach it.
> >So, if a teacher of young children doesn't teach with an eye on the
> >future, he or she is not doing the kids a favor.
> >
> >
>
> Frankly, I disagree entirely with this *we have lots of things to
> teach and not enough time to teach it* There is nothing that
> says that someone cannot take more years to learn since
> learning is lifelong.
>
> By pushing kids, we create stress instead of a love of learning
> and love of learning is what people need throughout life in order
> to push themselves to learn new things and to suceed.
>
> Aside from that not everyone is suited to a high pressure life
> and I am not sure that people who don't want to live that way
> aren't better off in many ways if they choose to live differently
> from your model of constant work.
>
While I admire your view, and wish it were true, I fear that it is not
realistic.
A child grows at a definite rate. At a certain point, the person ceased to
be a child and must become an adult. It is therefore necessary that the
person be equipped with the skills and knowledge needed to function as an
adult, or his/her future will be compromised. As our society becomes more
complex, the total amount of skills and knowledge has grown.
I fear that many, if not most, children will not acquire the necessary
knowledge and skills if we rely only on love of learning. Though most
children do have a love of learning, they also have many other loves:
activity, play, socializing, relaxing, being entertained. Some of these can
be used for learning, but not always and in all circumstances. I regret to
say that I believe that, in fact, the children do need to be pushed if they
are to perform well.
The analogy is sports. Most children love sports and try many of them.
Most of them will never master any but the most childish sports without
coaching, and without at least some pressure to improve. While it can be,
and too often is, overdone, most coaches do apply pressure on their young
athletes, and most young athletes benefit from it.
Jim Wayne
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On 2 Jul 2002 08:33:45 -0500, hru...@odds.stat.purdue.edu (Herman
>Rubin) wrote:
>>In article <5v63iusm4b9f580gj...@4ax.com>,
>>Alberto Moreira <junk...@moreira.mv.com> wrote:
>>>toto <scar...@wicked.witch> said:
..............
>>In the labs I have taken, we were told what experiments to
>>perform, and even what results to find. There was neither
>>direction nor discovery.
>And the experiments lead to discovery by the students unless
>you never had an *aha* moment when you did perform and
>experiment and saw that the result was as expected.
If someone has the mathematics needed to understand the
precise formulation, the *aha* moment comes from the
mathematical reasoning. How should an expected result
receive such a response?
And, of
>course, in high school physics the results of many labs don't
>actually come out the way they are supposed to for various reasons,
>so fudging becomes the norm and the students learn nothing from
>that at all.
The only labs I had were college physics (tested out of
high school physics, and was told to do that in chemistry)
and college biology. There were experiments which did
not work out, but I believe the main purpose of these labs
was to develop proficiency in handling the equipment.
The mathematics can, and should, be taught early.
Scientists today do not fiddle around that much in labs;
they work with mathematical models and data in computers.
>In elementary school, there is more leeway to do labs that encourage
>that *aha* moment in the student's mind.
Only if the student has been kept from using it logically.
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On Tue, 02 Jul 2002 09:00:30 -0400, Alberto Moreira
><junk...@moreira.mv.com> wrote:
>>toto <scar...@wicked.witch> said:
>>>Since neither of them teaches young children, it is hard
>>>to tell what they would do or if they could teach it to the
>>>average elementary school child (not the 10% who are
>>>talented already)
>>Children don't remain young forever, and the period where we can do
>>thing nice and soft is two or three years at most. After that, reality
>>must sink in: we have lots to teach and not enough time to teach it.
>>So, if a teacher of young children doesn't teach with an eye on the
>>future, he or she is not doing the kids a favor.
>>Alberto.
>Frankly, I disagree entirely with this *we have lots of things to
>teach and not enough time to teach it* There is nothing that
>says that someone cannot take more years to learn since
>learning is lifelong.
As I believe that teen-agers should know more than
college graduates now get.
>By pushing kids, we create stress instead of a love of learning
>and love of learning is what people need throughout life in order
>to push themselves to learn new things and to suceed.
If you want to create a love of learning, do not even
consider having a child in a dumbed down class for a week,
or have the teaching be by someone who tells the child how,
but refuses to discuss why.
Have them learning real subject matter as soon as they
can, and avoid having them just memorize and do routine
problems. Do not make school boring, or have them spend
their time in any ways other than real academics.
>Aside from that not everyone is suited to a high pressure life
>and I am not sure that people who don't want to live that way
>aren't better off in many ways if they choose to live differently
>from your model of constant work.
Who said it is constant work? Allow them to open their
minds, instead of doing what the educationists and
child psychologists insist is age-appropriate. As we
say in other newsgroups, your mileage may vary, and
trying to fit them into the mold of those who are unable
to comprehend this believe they should is nothing less
than severe child abuse.
Herman Rubin
Donna Metler <nospam_...@bellsouth.net> wrote:
>--
>Donna DeVore Metler
>Orff/ Band/Choral music, Lester Focused Literacy School
>Mother to Angel Brian Anthony, 01/01/02 (22 weeks, severe PE/HELLP syndrome)
>"Bob LeChevalier" <loj...@lojban.org> wrote in message
>news:e0v3iug2j1sk8c3vu...@4ax.com...
>> hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>> >In article <l773iuolek1oc1ain...@4ax.com>,
>> >Alberto Moreira <junk...@moreira.mv.com> wrote:
>> >>Bob LeChevalier <loj...@lojban.org> said:
...................
>> >The educationists have no idea how unimportant drill is.
>> And yet, in music, drill is critical and the difference between one level
>and
>> the next seems to at what level you do your drilling.
This is drilling in performance, not understanding.
Transferring this to
>> math, a person who is not drilling arithmetic facts may not progress to
>> algebra,
Only because the teachers cannot recognize that being able
to perform arithmetic, which is what that is, has nothing
to do with understanding arithmetic, let alone understanding
algebra. Basic algebraic ideas belong in first grade, as
they have no prerequisites other than an ability to understand
extremely simple grammar and vocabulary. They are a simple
language with little vocabulary. Then one can move on to
understanding mathematics; learning the multiplication tables
is not going to help, and it seems may hurt.
a person who drills on proving Euclidean theorems will be able to
>> move on to higher proofs.
If the proofs are done logically, not geometrically.
Learning mathematical logic, which has been taught to
fifth graders and should come earlier, will help more.
Drilling on it will not help much, as the proofs are
too simple, and use too little.
Proofs should be used very early, even to understand
numbers, which is rare among schoolteachers, and then
continued.
..............
>In math, though, you're constantly learning new skills, and what is learned
>in one branch doesn't always carry over to others.
It is not "skills" which are important, but concepts and
understanding. If it is understood, it can be used in
other situations. I question if what is learned in one
branch is "carried over" into another; it may be applied,
but that is different.
A student who fails
>algebra can still do quite well in Geometry, for example. Arithmetic isn't
>terribly relevant to a theoretical mathematician.
Very definitely. It is not much more relevant to an
"applied mathematician". For the user of mathematics,
it is rarely of much use, unless the person is at the
forefront of applying mathematical theory.
Bob LeChevalier
>>Mathematicians, and even students who learned mathematical
>>concepts in sound abstract courses, were able to teach the
>>"new math" quite well to the average elementary school child.
>
>The students they taught were not average elementary school
>students. This was done in University Lab schools which don't
>have average kids.
I t sounds like you actually know what he is talking about, toto. I've asked
him for years whether these efforts were ever documented (since I don't
believe his interpretation), but he never provided a cite or even a name of
someone involved in the program. It is hard to imagine that a noteworthy and
supposedly successful experiment was never written up.
>>Even if the tens of thousands of children on whom it was
>>tested were not quite representative, that should be
>>sufficient evidence that it can be done for at least a
>>large proportion.
>>
>Please site the 10s of thousands of kids taught by mathematicians
>who were not teachers? The first kids *may* have been taught
>by mathematicians, the next groups were taught by teachers who
>were trained by those mathematicians, not by the mathematicians
>themselves.
And of course, per Herman, that teaching supposedly failed because unlike the
kids, the teachers couldn't learn. But he dates this experiment long before
the New Math that I grew up with, which failed because the parents fought it
tooth and nail because it was not the math they knew to be math, and THEY
didn't understand it. I still remember my college educated mother wondering
what my sister's 4th or 5th grade textbook was talking about with its
"commutative" and "associative" properties. In my case they did not have the
textbooks yet at those grades, and I was probably 6th or 7th grade before I
learned the terms (and I couldn't figure out why they were worth teaching
special names for, since it was OBVIOUS that 3+4 = 4+3).
lojbab
Bob LeChevalier
>$20/hr is also extremely LOW for individual private lessons.
You're right, I must have been asleep when I wrote that. The place that has
beginning lessons charges $15 for a half hour (they charged $12 back when my
kids started), and my kids each had one year where they paid only $20 for an
hour, but it was a grad student in training doing the teaching. The
professional experienced teachers seem to charge around $20 per 1/2 hour or
$40 per hour.
>It is next to impossible to make a full-time job out of teaching privately,
>because most of your students are in school during the day, and there are
>only so many hours in the evening and on weekends. If you're not a strings
>teacher or piano teacher, it is very difficult to get enough students, too,
>because while there are hundreds of saxophonists (for example, since I am a
>sax teacher) in a good sized city, only a very small percentage of students
>take lessons privately, and once a student is serious enough to be planning
>to go to college, it is usually much better for the student to enroll in
>private lessons as a university/conservatory class, if at all possible, and
>get college credit and, more important, lessons with someone who has
>professional credentials which will mean more to universities or
>conservatories.
I didn't know this, and my daughter is starting lessons with the private
teacher who is the oboe professor at GMU, and the college kids get credit for
her lessons, so maybe she would get college credit as well. Of course she is
2 years away from being eligible to attend GMU. I wonder who to ask.
lojbab
Bob LeChevalier
>>The basic experiments (which fit Alberto's description that a lab should have
>>specific instructions and not be mere exploration) also seem to support the
>>ordinal concept as well, wherein you sequence the rods in terms of length,
>>and observe that each successive rod differs from the prior one by a unit
>>length equal to the shortest rod. This can be generalized also by building
>>sequences where the length differs by 2 units, etc.
>
>This is not ordinal, but cardinal. And it requires realizing
>that units combine in the counting manner.
>
>The ordinal approach requires that the comparison of one rod
>must be made with an initial segment of the other. How likely
>is it for a child to get this?
They are directed to it. The Cuisinaire book that I have had specific
directions for exercise and activities; what the kid should do, and what
questions to ask.
I noted that they are shown that the unit length is the shortest rod in the
set, and each succeeding rod in the ordered set differs from the preceding
one by that one unit, so I think that matches what you just said.
The main difference between what Alberto would want and what they do is that
Cuisinaire's lessons are meant for a teacher or parent who does NOT
understand the math. They have a supplement for those parents that do know
math, that explains notationally what is going on in each lessons, and that
supplement is written at the college level, IMO. But since Cuisinaire can be
started entirely prenotation (indeed before most kids can properly hold a
pencil), they do not build or require the notational stuff in the lessons.
The knowledgeable person can add that stuff in for the kids when they can
handle it.
>I cannot see how to present a
>solid foundational approach to the integers without using the
>ordinal structure at some time, but I do not believe that this
>was integrated into the "intuitive" approach of the new math.
>The ordinal approach is counting by succession, not by arbitrary
>assignment as in the cardinal approach.
The Cuisinaire exercises that I tried with my kids seemed to be based on
ordered succession, though I may be fuzzy since it has been several years.
lojbab
Seveigny
importantly, that your explanation didn't make any sense. Your posts state
that you had a superior understanding of mathematics because you taught
mathematics intensive courses. I consulted an expert in Mathematics who
disagreed with you as well. Since he teaches the same subject as you, one
might expect he would have a similar viewpoint, or at least understand your
perspective. He does not.
You can put out your views, whether they spring from your education or your
experiences. The post where you put forth detail was not compelling. Your
opinions are your own and you are entitled to them. However, they are
simply opinions. Dressing them up as fact is fallacious.
Cate
"Alberto Moreira" <junk...@moreira.mv.com> wrote in message
news:o093iu4sd093l9nub...@4ax.com...
> "Seveigny" <sarra...@attbi.com> said:
>
>
> >Good question. Alberto's condescending tone ticked me off but his
> >explanations didn't make sense either.
>
> teacher. In another post I went through a bit of detail to contrast my
> views to those you presented in your other post, though, although I
> normally try to go into those kinds of "t'is - t'aint" discussions.
>
> I have a view of things that grew up from my experience teaching them.
> If you don't like them, sorry, I can't change it, they came from the
> world as I keep seeing it.
>
>
> Alberto.
>
>
>
>
toto
Rubin) wrote:
>University Lab schools are not that great. They are not
>selective on the basis of ability.
Every one I have observed in has students who are above
average in general. They may not purposely select for this,
but the fact is that there is self-selection involved because
of the parents who choose such schools.
toto
wrote:
Many people are currently scaling back their lifestyle and they are
still functioning quite well. While some of that is forced by
corporate downsizing, the fact is that there are people who are
quite simply choosing to do things differently and who choose
quality of life over the life that strives so hard for material
gains that the quality of life suffers for it.
>A child grows at a definite rate. At a certain point, the person ceased to
>be a child and must become an adult. It is therefore necessary that the
>person be equipped with the skills and knowledge needed to function as an
>adult, or his/her future will be compromised. As our society becomes more
>complex, the total amount of skills and knowledge has grown.
>
Yes, but who says that this child needs college academics to
function perfectly well as an adult.
Academics provides only a particular kind of skills. Not everyone
needs that skill set and there are people who are perfectly happy
not to pursue them.
>I fear that many, if not most, children will not acquire the necessary
>knowledge and skills if we rely only on love of learning. Though most
>children do have a love of learning, they also have many other loves:
>activity, play, socializing, relaxing, being entertained. Some of these can
>be used for learning, but not always and in all circumstances. I regret to
>say that I believe that, in fact, the children do need to be pushed if they
>are to perform well.
>
Play, socialization, acitivity all involve learning and children
learn through all of those. Play for children is actually how they
learn most of what is important about the world.
If a child keeps the love of learning, then you don't need to push
him to learn because he will pursue learning. He might not pursue
it in the order or method that adults believe is necessary, but he
will pursue it. If adults become facilitators instead of trying to
control the direction of a child's learning, then the things that
child need will be mastered at the pace that child can handle the
learning.
>The analogy is sports. Most children love sports and try many of them.
>Most of them will never master any but the most childish sports without
>coaching, and without at least some pressure to improve. While it can be,
>and too often is, overdone, most coaches do apply pressure on their young
>athletes, and most young athletes benefit from it.
>
I disagree that most children *love* sports. Many children enjoy
them, but many more do not. This is especially so as the
competitive atmosphere takes over from the atmosphere of
playing the sport for its own sake.
Actually the very best coach my son had did not pressure the kids.
He was often heard to say *good play, lads* when the other team
played well as he often did with the boys on his own team. He
taught by example and love of the game. He put the drills and
skills out there and motivated all the kids to participate and work
to the best of their ability not by pressuring them, but by showing
them what they could do if they tried and wanted to do it. My son,
btw, would still be playing at some level but for bad knees. This
particular coach imparted his love of the game so that many of the
kids are coaching and still playing at 30 (not professionally though
I do think a few are playing or working with professional teams).
Dorothy
>Jim Wayne
toto
Rubin) wrote:
>As I believe that teen-agers should know more than
>college graduates now get.
More what? That's the problem. More academic knowledge?
Not every teenager wants to pursue mathematics or science.
Some will be carpenters. Some will be actors or artists or
musicians. Some will even be farmers or retail clerks in terms
of their careers.
toto
is at the beginning so that for agent, if I just hit reply,
it deletes all your the text below it?
At any rate, meet you further down.
I can see that it is. But I don't see how it would apply to
beginning mathematics students and that is what most high
school students are (much less elementary students).
I do think that even at the lower levels, it is important for
beginning students to be able to justify their reasoning. So
I can see a use for having a small class working on problems
that are enough different that they must clearly communicate
their methods to the rest of the students and be able to
*critique* the other students methods at least to a beginning
extent. Or for the students to pose meaninglul questions to
the student who is explaining.
Dorothy
Donna Metler
--
Donna DeVore Metler
Orff/ Band/Choral music, Lester Focused Literacy School
Mother to Angel Brian Anthony, 01/01/02 (22 weeks, severe PE/HELLP syndrome)
"toto" <scar...@wicked.witch> wrote in message
news:ujd5iu4i4smq3v7bh...@4ax.com...
> On 2 Jul 2002 16:58:02 -0500, hru...@odds.stat.purdue.edu (Herman
> Rubin) wrote:
>
> >University Lab schools are not that great. They are not
> >selective on the basis of ability.
>
> Every one I have observed in has students who are above
> average in general. They may not purposely select for this,
> but the fact is that there is self-selection involved because
> of the parents who choose such schools.
>
> Dorothy
Especially since most give priority to students of faculty, and university
faculty, by definition, are highly educated individuals.
Campus school here is officially part of the public school system, and is
easily the most difficult city elementary school to get into. If you are not
already part of the university community, you're probably not going to get
in. The University also sets criteria for teachers, and hires separately
from the district, and generally campus teachers are extremely well
qualified.
The music department at UM doesn't have their students do observation or
practicum at Campus, specificially because it is NOT a typical teaching
situation.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>toto <scar...@wicked.witch> wrote:
>>>Mathematicians, and even students who learned mathematical
>>>concepts in sound abstract courses, were able to teach the
>>>"new math" quite well to the average elementary school child.
>>The students they taught were not average elementary school
>>students. This was done in University Lab schools which don't
>>have average kids.
>I t sounds like you actually know what he is talking about, toto. I've asked
>him for years whether these efforts were ever documented (since I don't
>believe his interpretation), but he never provided a cite or even a name of
>someone involved in the program. It is hard to imagine that a noteworthy and
>supposedly successful experiment was never written up.
>>>Even if the tens of thousands of children on whom it was
>>>tested were not quite representative, that should be
>>>sufficient evidence that it can be done for at least a
>>>large proportion.
>>Please site the 10s of thousands of kids taught by mathematicians
>>who were not teachers? The first kids *may* have been taught
>>by mathematicians, the next groups were taught by teachers who
>>were trained by those mathematicians, not by the mathematicians
>>themselves.
I do not have the actual reports, but do you think the
educationists would have accepted a different method of
teaching mathematics if it had NOT been tested? The
program was started in the late 40s, from an observation
that a child who was adept in arithmetic had no idea of
what any of it meant; it was just procedures.
>And of course, per Herman, that teaching supposedly failed because unlike the
>kids, the teachers couldn't learn.
This was observed almost immediately. Major efforts were
made to upgrade the teachers. I do not know if they
couldn't learn, but that they couldn't get away from the
idea that, if they could do the arithmetic operations,
they knew what numbers were.
I was at a university where there was a summer institute
for high school teachers of mathematics. These were not
the strongest mathematically among high school teachers,
but also not at the weak end. One of those involved in
teaching it estimated the proportion who could manage to
learn, under any conditions, the concepts in the basic
rigorous undergraduate courses, what are the foundations
of analysis and algebra, was only about 10%.
But he dates this experiment long before
>the New Math that I grew up with, which failed because the parents fought it
>tooth and nail because it was not the math they knew to be math, and THEY
>didn't understand it.
Didn't anyone tell them that arithmetic is not mathematics,
and not at all basic in mathematics? That they did not
understand it is because they had never learned it, and
because they did not look into the materials to see what
was actually there. Going from methods to concepts is the
VERY hard part, and the more drill on the methods, the
harder. "I know how to add and multiply numbers; I don't
want you to tell me what it means" is essentially an
admission that it is only the mechanics which mean anything.
I still remember my college educated mother wondering
>what my sister's 4th or 5th grade textbook was talking about with its
>"commutative" and "associative" properties.
Think algebra first, and think of it as language. Those are
linguistic terms referring to mathematical objects. There is
no problem in teaching what these terms mean to kindergarten
children.
In my case they did not have the
>textbooks yet at those grades, and I was probably 6th or 7th grade before I
>learned the terms (and I couldn't figure out why they were worth teaching
>special names for, since it was OBVIOUS that 3+4 = 4+3).
It may be intuitively obvious after you know what "3", "+",
and "4" mean. But intuitively obvious does not make it
so. Mathematics is also not like the sciences, where one
acts as if something is true by generalization from
observation; that is only conjecture in mathematics. It
takes proof. The new math took one approach to defining
these, which I do not think best, and did not at the time,
but at least it defined them and provided proofs of the
general result, which is certainly NOT obvious until numbers
and addition are defined.
There is an old joke about a kid telling his parents, "The
teacher does not know her onions. Yesterday, she told us
that 4+2 = 6. Today, she told us that 5+1 = 6." It is
advisable not to use the word "obvious", except very carefully.
Good education is for the future; this approach needs to
be taken from the beginning, and a child put off by "you
know enough" or "you will learn it when you need to" is
being turned off from learning.
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On 2 Jul 2002 20:22:05 -0500, hru...@odds.stat.purdue.edu (Herman
>Rubin) wrote:
>>As I believe that teen-agers should know more than
>>college graduates now get.
>More what? That's the problem. More academic knowledge?
>Not every teenager wants to pursue mathematics or science.
>Some will be carpenters. Some will be actors or artists or
>musicians. Some will even be farmers or retail clerks in terms
>of their careers.
Change that to those who spend a comparable amount of time
to what they are now spending for those who want to learn.
At least 20% of the children could learn much more, and
understand better, by middle school, than they are now
allowed to get by the end of high school, even with an
honors program.
toto
<loj...@lojban.org> wrote:
>In my case they did not have the
>textbooks yet at those grades, and I was probably 6th or 7th grade before I
>learned the terms (and I couldn't figure out why they were worth teaching
>special names for, since it was OBVIOUS that 3+4 = 4+3).
>
>lojbab
Exactly what my kids thought, though they learned the vocabulary.
The idea that addition and multiplication were commutative and even
the distributive property already simply was obvious to them before
they ever learned the names for such things.
I don't think though that naming properties is what Herman thinks of
as the *new math.* I wish, in a way, that I could get hold of
something that showed exactly what the very first class contained
and who taught it and how he did so. I don't think that what has
been kept or even what was tried in *most* schools was what the
program started out as.
toto
<nospam_...@bellsouth.net> wrote:
>Donna DeVore Metler
>Orff/ Band/Choral music, Lester Focused Literacy School
>Mother to Angel Brian Anthony, 01/01/02 (22 weeks, severe PE/HELLP syndrome)
>
>"toto" <scar...@wicked.witch> wrote in message
>news:ujd5iu4i4smq3v7bh...@4ax.com...
>> On 2 Jul 2002 16:58:02 -0500, hru...@odds.stat.purdue.edu (Herman
>> Rubin) wrote:
>>
>> >University Lab schools are not that great. They are not
>> >selective on the basis of ability.
>>
>> Every one I have observed in has students who are above
>> average in general. They may not purposely select for this,
>> but the fact is that there is self-selection involved because
>> of the parents who choose such schools.
>>
>> Dorothy
>Especially since most give priority to students of faculty, and university
>faculty, by definition, are highly educated individuals.
>
I almost said that too, but since I don't have a base that tells me
that all lab schools do that I admitted it. I was pretty sure that
was the case too.
>Campus school here is officially part of the public school system, and is
>easily the most difficult city elementary school to get into. If you are not
>already part of the university community, you're probably not going to get
>in. The University also sets criteria for teachers, and hires separately
>from the district, and generally campus teachers are extremely well
>qualified.
>
>The music department at UM doesn't have their students do observation or
>practicum at Campus, specificially because it is NOT a typical teaching
>situation.
Yep.. The lab schools here that I have observed are also untypical
because classes are smaller and often experimental programs are
being tried.
Dorothy
toto
Rubin) wrote:
>In article <gge5iu8h2ja1vlpcp...@4ax.com>,
>toto <scar...@wicked.witch> wrote:
>>On 2 Jul 2002 20:22:05 -0500, hru...@odds.stat.purdue.edu (Herman
>>Rubin) wrote:
>
>>>As I believe that teen-agers should know more than
>>>college graduates now get.
>
>>More what? That's the problem. More academic knowledge?
>>Not every teenager wants to pursue mathematics or science.
>
>>Some will be carpenters. Some will be actors or artists or
>>musicians. Some will even be farmers or retail clerks in terms
>>of their careers.
>
>Change that to those who spend a comparable amount of time
>to what they are now spending for those who want to learn.
>At least 20% of the children could learn much more, and
>understand better, by middle school, than they are now
>allowed to get by the end of high school, even with an
>honors program.
Now, we can agree to an extent. No child should ever be held
back from learning more.. But, what you forget is that no child
should be hounded into learning more either. We must begin
to go by the student's innate desire to learn. We don't try to make
an infant walk before he is developmentally ready. Why must we
try to make a child read or write before his readiness kicks in?
I agree that we must be careful to look for signs of readiness,
present material in ways that scaffold the child higher and higher
and get out of his way if he is ready and willing to learn. I do
quarrel with teachers who refuse to present things to children
because they believe its not *developmentally appropriate* and
I also find it disheartening that some teachers of young children
don't understand the ideas behind mathematics and so do
misteach. But I do not think this is a majority of those teaching
as you do. And the structure of schools *must* change if we are
going to be able to get to individually paced instruction.
I would agree that multi-aging classrooms makes sense and that
smaller classes make sense and that elementary teachers need
to be specialized more as well. I see these things coming down
the pike slowly but surely if we don't simply destroy the entire
system first by bad political decisions. I'm not optimistic at all
that we will have *any* education for the kids who are not gifted by
the time the political questions are answered. And since I care
that all students get an education to the highest extent of their
individual abilities, I think that leaving those who are not able to
do the kind of academics *you* want is a scandal.
Herman Rubin
Alberto Moreira <junk...@moreira.mv.com> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) said:
>>This is drilling in performance, not understanding.
>The problem is, merely understanding is not enough.
>>Only because the teachers cannot recognize that being able
>>to perform arithmetic, which is what that is, has nothing
>>to do with understanding arithmetic, let alone understanding
>>algebra. Basic algebraic ideas belong in first grade, as
>>they have no prerequisites other than an ability to understand
>>extremely simple grammar and vocabulary. They are a simple
>>language with little vocabulary. Then one can move on to
>>understanding mathematics; learning the multiplication tables
>>is not going to help, and it seems may hurt.
>Again: understanding isn't enough. We need the drill because we need
>the ability to manipulate.
To set things straight, I am quite good at arithmetic, but
it still does nothing except save time.
I see no difference between doing the arithmetic by hand,
or having it done on a computer or calculator.
And I would contend that at this level,
>those "basic algebraic ideas" don't exist - algebra is merely
>arithmetic in disguise.
Definitely not. The most important part of algebra is as a
language to translate from the "real world" problem to the
problem in formal mathematics. Those who work in physics,
chemistry, biology, economics, agriculture, or anything
else need THIS more than any ability to grind out answers
themselves. Most real world problems are of this nature,
that even someone who is quick and accurate in the arithmetic
operations is not going to be able to answer them in "real time".
I have done numerical work manually, but that is mainly because
decent computer facilities have not existed, and do not exist.
Most of the packages do what the package produces like easily,
but everything else becomes harder than starting over.
I started out doing mathematics for econometricians. There is
no way that the number sense and the ability to calculate,
which they had, was useful. It was their ability to formulate,
and to translate back, which was.
toto
<loj...@lojban.org> wrote:
>I t sounds like you actually know what he is talking about, toto. I've asked
>him for years whether these efforts were ever documented (since I don't
>believe his interpretation), but he never provided a cite or even a name of
>someone involved in the program. It is hard to imagine that a noteworthy and
>supposedly successful experiment was never written up.
It was written up, I am sure. I can give you some cites about who
was involved having found an interesting article online that does
take Herman's view of it's failure in many ways. The article admits
to bias against teachers educated in schools of education, but it
does give a historical note about who started it and how it was
done.
http://www.math.rochester.edu/u/rarm/smsg.html
:The year 1958 therefore kicked off the largest and best
:financed single reform effort ever seen in mathematics
:education, the School Mathematics Study Group (SMSG),
:upon which the National Science Foundation (NSF) spent
:millions of dollars over its twelve-year lifetime.
:
: Edward Begle, a professor of mathematics at Yale University,
:was chosen to head the new organization, and gave up topology for
:this new and unfamiliar calling. The existing professional educa-
:tional bureaucracy, later called "the PEB" by William Duren, a
:reforming mathematician of the time, was thus suddenly outflanked
:by a new party.
****
:To put first things first, he assembled several separate teams
:of mathematicians to write exemplary textbooks, eventually
:covering all grades 1-12 and a bit more, that would be free of
:the ignorance, ambiguity, opacity, irrelevance and tedium of
:the traditional curriculum. He included practicing schoolteachers
:in each writing team, hoping (vainly as it turned out) to keep his
:textbooks within the realm of the classroom possible; but the
:mathematicians drove the effort. SMSG invited all commercial
:publishers to study, copy, or plagiarize these texts, which SMSG
:placed in the public domain as models, freely.
:Simultaneously, SMSG established hundreds of Institutes,
:i.e. special college courses for existing teachers, some in the
:summers and some on Saturdays, to which eventually thousands (paid
:by the NSF) came to study the new material, to practice its
:pedagogy under the eyes of SMSG authors and master teachers, and
:then to carry the books back into the world for classroom testing
:on a nation-wide scale. The writing groups would reassemble
:summer after summer, study the reports from the field, and revise
:the texts and the teacher's guides for the next set of Institutes
:and experimental classes.
:
: Almost half the nation's high school teachers of mathematics
:attended at least one such Institute during the 12 year life of
:SMSG; but an equivalent seeding was impossible for elementary
:school teachers, who outnumbered the high school math teachers ten
:to one. While there were some Institutes for elementary school
:teachers, these were mainly for experimentation. The SMSG books
:themselves achieved unexpectedly wide circulation, and were
:indeed, as Begle had urged, enthusiastically if often ignorantly
:imitated, even (or especially) at the more elementary levels. And
:the research literature produced in the colleges of education, and
:the journals of classroom practice written and read by teachers,
:were for the entire decade of the sixties dominated by obeisance
:to the SMSG program.
:
: The result, after twelve years, was total failure. By any
:reasonable measure, and measures were taken, school mathematics
:was worse off in 1975 than it had been in 1955. The idiocies of
:the older curriculum had in most places been removed, but often to
:be replaced with new ones.
Of course it failed. Since almost no elementary teachers were
included, it could not have been successful. Top down doesn't work
very well. Programs should have been introduced bottom up, imo..
Had these people included elementary school teachers *first* things
might have been different.
Another note: I don't believe that almost half of all high school
teachers were included in such institutes either, but that's
something I cannot verify.
******
:To take an example, the language of the "theory of sets" has
:been basic among mathematicians for a hundred years, and can ease
:enormously the path to much that people find perplexing in school.
:Anyone should be able to learn enough about sets and this vocabu-
:lary in a very few hours to permit him in consequence to under-
:stand an honestly presented course of high school mathematics
:including all the traditional material and more; his savings in
:time will have exceeded those few hours a hundredfold, and in
:understanding immeasurably. SMSG introduced set theory into its
:first books, which as it happened were for the high school level.
:Later books, written for grade-school years, also introduced the
:subject of sets, hoping later to make use of it when revised high
:school books were written. It therefore turned out that for a
:time -- all the time SMSG had, alas, in its short career -- a
:chapter on sets appeared at the opening of every year's textbook,
:unfortunately making it appear as if sets were the be-all and end-
:all of Newmath. This redundancy was copied into the commercial
:texts of the time as well, and teachers leaped on it to the
:neglect of more prosaic matters, like getting a correct answer in
:arithmetic.
:
: Easy as it looked, teachers didn't always get the notion of
:"set" straight themselves, and could teach the most egregious
:confusions as truth. One textbook lesson plan suggested that the
:teacher, as an example, distinguish the subset "boys" from the
:subset "girls" (in the set "this class") by asking the boys to
:stand, and then the girls to stand, and so on; one teacher I heard
:about then asked "the set of boys" to stand up. But while boys,
:being human, can stand, sets cannot. So fine a distinction may be
:meaningless to a third-grade teacher, or to anyone who has never
:made real use of it; but if exactly that distinction is not made
:plain, and into a habit of mind and speech, the notion of set is
:valueless in later mathematical reasoning.
:
This one is interesting and somewhat confusing. While I certainly
understand the distinction being made here, I am not sure how
this particular error will contribute to later misunderstandings by
the students. And this could easily be remedied with the
elementary teachers by making the distinction plain to them
in workshops.
: On the other hand, SMSG and its imitators were also guilty
:of some pointless pedantry, ridiculous even if logically correct:
:"Write the numeral that names the number solving 3x -7 = 8," for
:example. That's not even English. If you actually ask a mathema-
:tician to write down his phone number he will cheerfully hand you
:a numeral without a moment's hesitation or apology. He can make
:the distinction, sure, but he only does it when it counts.
:
And this is important too. The distinctions in terms of vocabulary
really only count in certain situations.
:Towards the end, Begle wrote, "I see little hope for any
:further substantial improvements in mathematics education until we
:turn mathematics education into an experimental science, until we
:abandon our reliance on philosophical discussion based on dubious
:assumptions, and instead follow a carefully constructed pattern of
:observation and speculation, the pattern so successfully employed
:by the physical and natural scientists." Begle himself died a
:disappointed man six years later, though he had continued after
:SMSG to work brilliantly towards a proper study of mathematics
:education. His disappointment was for the future more than for
:SMSG, because he foresaw correctly that PEB-sponsored research in
:education would not follow his sensible, if unexciting, prescription.
I think that his reasoning about *why* thinks failed is specious at
best, but the article provides an interesting perspective on how
things were done.
Herman Rubin
toto <scar...@wicked.witch> wrote:
I was going to send private email, but the address is not
present.
>On Wed, 03 Jul 2002 03:49:24 GMT, Bob LeChevalier
><loj...@lojban.org> wrote:
>>In my case they did not have the
>>textbooks yet at those grades, and I was probably 6th or 7th grade before I
>>learned the terms (and I couldn't figure out why they were worth teaching
>>special names for, since it was OBVIOUS that 3+4 = 4+3).
>>lojbab
>Exactly what my kids thought, though they learned the vocabulary.
>The idea that addition and multiplication were commutative and even
>the distributive property already simply was obvious to them before
>they ever learned the names for such things.
Lots of things are obvious which are not so. One of the
important points in mathematics is that it is necessary to
at least for someone to be able to prove something, not
just to have it obvious. It is not necessary for
understanding something to be able to prove it.
I do not know how much, if any, statistics you know, but
the very important Neyman-Pearson Lemma is conceptually
accessible to those with a high school algebra level, but
this understanding is rarely taught at any level, including
the highest. It is proved, usually in a special case, in
most mathematical statistics courses, with no insight
whatever into why it holds.
>I don't think though that naming properties is what Herman thinks of
>as the *new math.*
You are quite right. However, the properties are important
and they should be known, together with examples of situations
where they do not always hold. I cannot think of any easier
words to describe them.
I wish, in a way, that I could get hold of
>something that showed exactly what the very first class contained
>and who taught it and how he did so. I don't think that what has
>been kept or even what was tried in *most* schools was what the
>program started out as.
It might very well have been kept somewhere. I am not
familiar with education literature.
You could look at the serious of books produced by Suppes
in the late 50s or early 60s to find something intended
from first grade on. It was intended to be easier to teach,
but without losing the essence.
This was definitely not the early material, but AFAIK these
are still available.
~
~
~
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On 3 Jul 2002 10:34:08 -0500, hru...@odds.stat.purdue.edu (Herman
>Rubin) wrote:
>>In article <gge5iu8h2ja1vlpcp...@4ax.com>,
>>toto <scar...@wicked.witch> wrote:
>>>On 2 Jul 2002 20:22:05 -0500, hru...@odds.stat.purdue.edu (Herman
>>>Rubin) wrote:
>>>>As I believe that teen-agers should know more than
>>>>college graduates now get.
>>>More what? That's the problem. More academic knowledge?
>>>Not every teenager wants to pursue mathematics or science.
>>>Some will be carpenters. Some will be actors or artists or
>>>musicians. Some will even be farmers or retail clerks in terms
>>>of their careers.
>>Change that to those who spend a comparable amount of time
>>to what they are now spending for those who want to learn.
>>At least 20% of the children could learn much more, and
>>understand better, by middle school, than they are now
>>allowed to get by the end of high school, even with an
>>honors program.
>Now, we can agree to an extent. No child should ever be held
>back from learning more.. But, what you forget is that no child
>should be hounded into learning more either.
Sometimes, it is important to do this. When we taught our
children to read, we considered it important that they know
how do do so reasonably well before being exposed to the
whole word method, in vogue at the time. With our son,
the only problem was physical, in being able to combine the
sounds, but was not mental, and he was reading at a high
level before starting school. With our daughter, is was
a struggle; when she started school, which claimed to be
using a phonics approach, she thanked us for making her
learn a much stronger approach than that of the school.
Both of our children learned mathematics early, but not as
I would teach it now. Our son never took a mathematics
course to learn below the upper division university abstract
courses, and our daughter not below the high school level.
Her learning of mathematical logic was much later than that
of our son, but early enough that abstract mathematics was
quite natural. The material we used was partly intended
for fifth graders, and partly "more advanced".
We must begin
>to go by the student's innate desire to learn. We don't try to make
>an infant walk before he is developmentally ready. Why must we
>try to make a child read or write before his readiness kicks in?
How do you find out about readiness? You try to teach, and do
not give up on the first failure.
>I agree that we must be careful to look for signs of readiness,
>present material in ways that scaffold the child higher and higher
>and get out of his way if he is ready and willing to learn.
There are times when it is necessary to make a full transition.
One can claim that reading is fostered by reading to the child,
but this is not obvious. This will not work for any use of
variables, although it can be approached; I have posted first
grade versions using no "mathematical" concepts whatever.
I believe in direct instruction for concepts, with illustration
AFTER the concept is presented. Otherwise, the special cases
get confused with the concept, so lots of different types of
examples should be presented if at all possible.
I do
>quarrel with teachers who refuse to present things to children
>because they believe its not *developmentally appropriate* and
>I also find it disheartening that some teachers of young children
>don't understand the ideas behind mathematics and so do
>misteach. But I do not think this is a majority of those teaching
>as you do.
Unfortunately, I have far too much information to think otherwise.
And the structure of schools *must* change if we are
>going to be able to get to individually paced instruction.
To do this, learning must replace age grouping.
>I would agree that multi-aging classrooms makes sense and that
>smaller classes make sense and that elementary teachers need
>to be specialized more as well. I see these things coming down
>the pike slowly but surely if we don't simply destroy the entire
>system first by bad political decisions. I'm not optimistic at all
>that we will have *any* education for the kids who are not gifted by
>the time the political questions are answered.
I see no hope for the gifted in the present schools. I have
recently received an email in response to my posting on this
topic in another newsgroup which is from someone at least close
to gifted who became a high school dropout from the mistreatment.
He is now considering going to college, but does not have the
funds to do so.
And since I care
>that all students get an education to the highest extent of their
>individual abilities, I think that leaving those who are not able to
>do the kind of academics *you* want is a scandal.
I agree. And many of them will have to take longer, and get
less, than what is now essentially demanded. It takes more
than multi-aged classrooms; some must go slower. I see little
value in retention in most cases; teaching the same material
again the same way is not likely to do much better. Nor is
skipping quite adequate; we will need to use either highly
accelerated classes, or a much higher use of self study, or
electronic classes, but a 6 year old who can handle a high
school algebra class is likely to do it much faster.
Bob LeChevalier
>I do not have the actual reports, but do you think the
>educationists would have accepted a different method of
>teaching mathematics if it had NOT been tested? The
>program was started in the late 40s, from an observation
>that a child who was adept in arithmetic had no idea of
>what any of it meant; it was just procedures.
But no one but you has ever heard of any such research in the 40s. The
literature most people know about dates from the Sputnik panic, when people
tried to redesign mathematical education largely WITHOUT testing. If your
research actually was conducted in the 40s, it seems to have vanished without
a trace.
>>And of course, per Herman, that teaching supposedly failed because unlike the
>>kids, the teachers couldn't learn.
>
>This was observed almost immediately. Major efforts were
>made to upgrade the teachers. I do not know if they
>couldn't learn, but that they couldn't get away from the
>idea that, if they could do the arithmetic operations,
>they knew what numbers were.
>
>I was at a university where there was a summer institute
>for high school teachers of mathematics. These were not
>the strongest mathematically among high school teachers,
>but also not at the weak end. One of those involved in
>teaching it estimated the proportion who could manage to
>learn, under any conditions, the concepts in the basic
>rigorous undergraduate courses, what are the foundations
>of analysis and algebra, was only about 10%.
Why would someone expect that most people could learn a couple of years worth
of undergraduate math in a summer institute? Methinks it is necessary to
figure out which of those concepts are truly IMPORTANT in order to be able to
teach what teachers actually are being called upon to do in the curriculum,
and then to hide the remainder under a "you probably don't need to understand
why, though it would be nice, but teach it THIS way". Of course the way that
is taught probably should be developed by the 10% that were both skilled
teachers and mathematically adept.
I've mentioned the Cuisinaire rods in this thread in part BECAUSE the people
who developed it were able to come up with a way that someone not
understanding the concepts could direct activities that DID teach the
concepts. The activities do the teaching and not the teacher.
> But he dates this experiment long before
>>the New Math that I grew up with, which failed because the parents fought it
>>tooth and nail because it was not the math they knew to be math, and THEY
>>didn't understand it.
>
>Didn't anyone tell them that arithmetic is not mathematics,
>and not at all basic in mathematics?
I'm sure they did, but just as you tell us, we don't believe you. We think
that we learned math in school, in my case I studied math in college, and
arithmetic was basic to almost everything I did in my college math classes
taught by the math department. Perhaps not basic in the sense that you mean,
but doing a math problem and even many kinds of proofs without knowing
arithmetic is pretty near impossible.
The real problem is that people learn math to solve problems, and you (and
Alberto) consider math to be a thing unto itself, independent of the problems
it is used to solve. That is a stark contradiction in philosophy that no
teaching will overcome. People want to learn what they can practically
apply, and see no reason to put a priority on learning other things.
You seem to think that this way of thinking is taught in schools, but I think
that this way of thinking is taught by life, and relatively few people will
consider theory more important than practice BY EXPERIENCE.
>Going from methods to concepts is the
>VERY hard part, and the more drill on the methods, the
>harder. "I know how to add and multiply numbers; I don't
>want you to tell me what it means" is essentially an
>admission that it is only the mechanics which mean anything.
As Alberto keeps saying, the proof of understanding is in demonstrating what
you can do with it. There is very little that you can do with the theory
that "means anything" to most people.
>> I still remember my college educated mother wondering
>>what my sister's 4th or 5th grade textbook was talking about with its
>>"commutative" and "associative" properties.
>
>Think algebra first, and think of it as language. Those are
>linguistic terms referring to mathematical objects. There is
>no problem in teaching what these terms mean to kindergarten
>children.
They can't teach linguistic terminology to kindergarten kids either, and I
have met college graduates who speak properly, but cannot remember what a
verb or a noun or an adjective is, and certainly don't know why they should
care what they are.
> In my case they did not have the
>>textbooks yet at those grades, and I was probably 6th or 7th grade before I
>>learned the terms (and I couldn't figure out why they were worth teaching
>>special names for, since it was OBVIOUS that 3+4 = 4+3).
>
>It may be intuitively obvious after you know what "3", "+",
>and "4" mean.
No. They are intuitively obvious because they WORK - they get the right
(i.e. useful) answer. (One reason I found complex numbers difficult was that
it took me a long time to see how imaginary i could possibly be "useful" even
if it was "correct" as an answer to a mathematical procedure. What I
"learned" was that higher math was a bunch of nonsense totally separated from
anything practical, and then only later learned that there WAS something
practical to it.)
The things that taught me that concepts in higher math were worth learning
came way too late, usually the year after I was supposed to have learned the
concept. Exceptions for me included the "birthday problem" (chances of 2
people sharing a birthday in a class of N people) which showed how intuitions
could be incredibly wrong but accurately predicted by probability theory,
writing a program to invert a Hilbert's matrix, which showed that rounding
errors at the umpteenth decimal place in computer math could amount to truly
enormous errors when propagated, and having to manually calculate something
in astrophysics (I recall it took some 10 hours with a calculator) and then
doing it trivially with a Fourier analysis.
Now maybe I shouldn't NEED such demonstrations of practicality, but I did,
and I was one of those supposedly gifted in math, who could not make the jump
to theory. And it wasn't directly because of what I was taught in school,
because most of the time I was self-teaching myself math by reading the
textbooks and doing the exercises on my own.
>Good education is for the future; this approach needs to
>be taken from the beginning, and a child put off by "you
>know enough" or "you will learn it when you need to" is
>being turned off from learning.
I heard those lines more in college than I did in the lower grades.
lojbab
Herman Rubin
toto <scar...@wicked.witch> wrote:
<On Wed, 03 Jul 2002 07:35:16 -0400, Alberto Moreira
<<junk...@moreira.mv.com> wrote:
>>I see things a lot more simple than that. Arithmetic is about
<>counting, there's no need for any further concept: the rest is, what
>>do you know, drill and experience. Our own fingers are well superior
>>to rods, and if I had to pick one prop to extend my students' fingers,
>>an abacus would be it.
>You might be interested in this article, Alberto
<http://supermath.com/chroniclenews2002.htm
<:The idea is not to memorize tables, as is the case with the
<:traditional Western way -- or "non-abacus way" as Watanabe
<:prefers -- math is taught. Rather, the idea is to let the abacus
<:and the brain do the work without the help of a preconceived
<:answer, until the brain can operate with only a "mental" abacus.
<:Many of the students continue to flick their fingers against a
<:table or the air as they manipulate imaginary abacuses. That,
<:says Watanabe, is the right brain hard at work, using imagination
<:as a tool.
<:"It's more about the thought process than anything else,"
<:Watanabe said. "It develops good fundamental skills. I can type
<:100 words a minute in Japanese. I quickly grasp computer
<:programs, and I've never had to take classes." And she gives
<:anzan and the abacus all the credit.
I agree. The abacus adds by counting, as is done in the
ordinal approach, and multiplies by repeated addition.
Bob LeChevalier
toto <scar...@wicked.witch> wrote:
>On Wed, 03 Jul 2002 03:49:24 GMT, Bob LeChevalier
><loj...@lojban.org> wrote:
>>I t sounds like you actually know what he is talking about, toto. I've asked
>>him for years whether these efforts were ever documented (since I don't
>>believe his interpretation), but he never provided a cite or even a name of
>>someone involved in the program. It is hard to imagine that a noteworthy and
>>supposedly successful experiment was never written up.
>
>It was written up, I am sure. I can give you some cites about who
>was involved having found an interesting article online that does
>take Herman's view of it's failure in many ways.
Actually the article, other than its trite labeling of the PEB as an
alternative to Herman's "educationists", seems to largely take the side of
Herman's opponents on why New math failed. To be specific:
>He included practicing schoolteachers in each writing team,
>hoping (vainly as it turned out) to keep his textbooks within the
>realm of the classroom possible; but the mathematicians drove the
>effort.
The textbooks were not practical for the classroom (I know because I studied
from the SMSG Geometry text, and also tutored other kids, some quite bright,
using it).
> Almost half the nation's high school teachers of mathematics
>attended at least one such Institute during the 12 year life of
>SMSG; but an equivalent seeding was impossible for elementary
>school teachers, who outnumbered the high school math teachers ten
>to one.
>SMSG introduced set theory into its
>first books, which as it happened were for the high school level.
>Later books, written for grade-school years, also introduced the
>subject of sets, hoping later to make use of it when revised high
>school books were written.
Elementary teachers largely did not get the training, and apparently the
elementary grade texts, written later, were never integrated into the
sequence (and I don't recall ever hearing that they had been written,
suggesting that they weren't even much tried. The SMSG only lasted 12 years,
and probably died before most schools would have even considered adopting
their books, given that textbooks are seldom replaced more than every several
years.)
>Tom Lehrer's 1965 song New Math, lam-
>pooning the pretentious language used to justify an inability to
>calculate, had the mathematical community itself laughing at the
>follies committed in the name of promoting a better understanding
>of mathematics.
Even mathematicians apparently felt that the ability to calculate was still
important.
> On the other hand, SMSG and its imitators were also guilty
>of some pointless pedantry, ridiculous even if logically correct:
>"Write the numeral that names the number solving 3x -7 = 8," for
>example.
Much of Herman's argument seems like just such pointless pedantry - it still
remains unclear what he WOULD have teachers teach, and how.
>Had SMSG really been tried? The mass of American tea-
>chers -- and children -- were not in the end exposed to, let alone
>taught, what the SMSG mathematicians prescribed.
Teachers cannot have failed to learn what they were never exposed to.
> Towards the end, Begle wrote, "I see little hope for any
>further substantial improvements in mathematics education until we
>turn mathematics education into an experimental science, until we
>abandon our reliance on philosophical discussion based on dubious
>assumptions, and instead follow a carefully constructed pattern of
>observation and speculation, the pattern so successfully employed
>by the physical and natural scientists."
Other than Herman's implication that such experimentation is only possible
with free market choice, it seems to me that Herman is the primary exponent
on these forums of philosophical discussion based on dubious assumptions. He
seems to have made no effort to look at what experiments HAVE been conducted
since then, much less considered whether they were carefully constructed.
>The article admits
>to bias against teachers educated in schools of education, but it
>does give a historical note about who started it and how it was
>done.
Yes! For the first time, I had a NAME, and it was trivial to use a search to
come up with a couple more and a little more history. (more on this in a
moment) But Begle started in 1958 and moved from Yale to Stanford in 1961.
This does not cohere with supposed research done at Stanford in the 1940s.
Meanwhile, a research program started at Stanford in 1961 could hardly be
expected to be completed, written up, and moved from the laboratory schools
to the pragmatics of regular schools in less than 10 years, and it was 10
years later that SMSG died.
So here is what I've found out quickly:
http://www.cah.utexas.edu/guides/mathematics.html
indicates that there are 130 feet of documentary records of the SMSG project
in an archive at the University of Texas, alone comprising around 20% of the
American mathematical historical archives collection there.
http://www.brook.edu/gs/brown/papers/loveless_pom.pdf
has another analysis of the failures of New Math. It gives some history, and
mentions some predecessor projects.
Looking for the references here, the earliest of these is summarized in
another site (cite below):
>In 1951, with the leadership of Max Beberman, the University of Illinois Committee on School Mathematics (UICSM) initiated a reform of the secondary school mathematics curriculum.
Well, that doesn't sound like elementary education, but the same article
discusses:
>What is sometimes referred to as the “Golden Age” of science and mathematics education began in the 1950s with development of new programs that eventually became known by their acronyms.
>...
>
>In mathematics the new programs included the University of Illinois Committee on School Mathematics (UICSM), the School Mathematics Study Group (SMSG), the Greater Cleveland Mathematics (GCM), the University of Illinois Arithmetic Project, the University of Maryland Mathematics Project (UMMP), the Suppes Experimental Project in the Teaching of Elementary-School Mathematics, and the Madison Project.
Finally we see a reference to Suppes and Stanford, but this was in the 50s,
not the 40s.
Using Beberman's name finally gave me a good bibliography of papers written
in that era discussing research in math education reform. A proposed course
on teaching of algebraic concepts (aimed at the level of preparing high
school teachers, alas for Herman) can be found at:
http://www.buffalostate.edu/~math/Grad/med605.html
And finally, I found a real gem, a collection of papers on the legacy of
Sputnik in science and math education.
http://www.nas.edu/sputnik/papers.htm
Three papers were specifically on math education:
http://www.nas.edu/sputnik/bybee1.htm
from which I took the quotes above on the Golden Age of Science and Math
education, and which mentioned Suppes. A couple other nice quotes:
>First, such educational criticism was not new, for example, in the late 1800s critics said that students were being “spoon-fed,” the curriculum was too easy, and music and art took too much time from fundamentals.
>Mathematics presented a different situation. Mathematicians criticized the new programs because the content was too abstract and neglected significant applications; teachers criticized the programs because they were too difficult to teach; and, parents criticized the new math because they worried that their children would not develop fundamental computational skills.
>Fifth, restricting initiatives to curriculum for specific groups of students, i.e., science and mathematically prone and college-bound students, resulted in criticism of Sputnik-era reforms as inappropriate for other students such as the average and the disadvantaged. To the degree school systems implemented the new programs teachers found that the materials were inappropriate for some populations of students and too difficult for others. Restricting policies or targeting programs opens the door to criticism on the grounds of equity. Proposing initiatives for ALL students also often results in criticism from both those who maintain there is a need for a specific program for those inclined toward science and mathematics and those who argue that programs for all discriminate against the disadvantaged.
http://www.nas.edu/sputnik/kilpatin.htm
which includes the following:
>Lesson Three: Teachers' knowledge is more easily changed than their teaching.
>
>The new math era is often criticized as a time when teachers were neglected and only the production of curriculum materials counted. Forgotten are the many institutes and courses that were provided to help teachers acquire the mathematical knowledge that the new materials demanded. For the most part, however, the courses were essentially college mathematics courses retooled for teachers. They provided new content knowledge but did not address the pedagogical problems of teaching that new content. Moreover, almost no attempt was made to deal with the conditions under which teachers work that inhibit their ability to change their teaching.
>
http://www.nas.edu/sputnik/lappan1.htm
From which I took the following quotes, which back my repeated assertion that
mathematics education reform has been poorly sold to parents:
>Public support for the direction of change is critical for success. Parents have to know how to help their young children with their mathematics homework and to feel secure. One lesson learned is that parents care deeply about education and want the best for their children. We laugh about parents demanding that their children suffer mathematics in the same way that they did. But we should be hearing the genuine desire to help their children that is underneath their words. When parents cannot understand the problems that their student is being asked to work on at home, they immediately assume that something is wrong.
>The way in which basic skills are attended to in innovative curricula must be clearly spelled out so that parents and administrators are satisfied that students will not be harmed. Each project must gather evidence that students are performing at an acceptable level on basic skills. The tradition that arithmetic has a place in the curricula because of the need to develop basic skills for trade and managing one’s affairs is very strong. Reform is in peril when parents and administrators are not satisfied that basic skills are a part of reform curricula. The NCTM Standards documents have been greatly misinterpreted in this area. By trying to move toward balance among conceptual development, problem solving, and skill development, the documents open themselves up to the interpretation that kids do not need to learn their “facts.” The real message is that estimation and mental arithmetic are more important than ever! In a technology environment a sense of the size of a number that is
expected as a result of a computation is essential to monitor the reasonableness of results.
Best of all was a summary history of 20th century math education at:
http://www.csun.edu/~vcmth00m/AHistory.html
This history suggests that Herman's constant focus on Dewey as the cause for
the decline in math education is misplaced, and he should be reserving his
vitriol for Kilpatrick, who studied under Dewey (but I've read somewhere that
Dewey rejected his positions).
The history also confirms my doubts that Herman is wrong about New Math
starting in the 40s:
>The University of Illinois Committee on School Mathematics headed by Max Beberman began in 1951 and was the first major project associated with the New Math era. Beberman's group published a series of high school math textbooks, and drew financial support from the Carnegie Corporation and the U.S. Office of Education.
And finally, researching Suppes, I found that his work largely started in the
late 60s and was more involved in computer-aided instruction than in teaching
algebra and logic to elementary kids. But I did finally find a reference to
his teaching of logic to elementary kids. Suppes wrote a paper in the mid
70s on teaching logic to elementary kids. It can be found at
http://www-epgy.stanford.edu/overview/research/homeeducation.pdf
and the home education emphasis in the paper suggests that this is indeed the
research Herman refers to, since he often discusses the gifted home education
list.
The experiment was performed using all of 28 children with a mean IQ of 170,
and only 4 kids were included with IQs less than 165 (the lowest IQ kid was
132, and he dropped out early). Their mean age was 12, the youngest boy was
11, though the youngest girl was 8 1/2. Of these 28, only 3 boys made quick
and steady progress, 2 of whom seemed motivated by the idea of understanding
the computer teaching system, 1 who seemed motivated by learning the
logic/algebra. 8 others moved more slowly but without the need for proctor
"pushing". The remaining children worked slowly and irregularly and needed a
lot of pushing to make progress, and 10 of the kids dropped out, though some
of these had completed part of the course.
Suppes' conclusions dispute Herman's claims for such programs, as well as for
the adequacy of computer-based instruction for even most gifted students,
much less the average ones:
>Second, on the basis of the high dropout rate experienced, we would conclude that a sustained program of home study for gifted students would require the introduction of considerable structure and also, probably, clear arrangements about academic credit for the course work done.
and
>The extensive data in correspondence coursesand the restricted but detailed data from the group of very gifted students described in the present section indicate that, in the case of home-based instruction, the dropout rate is a more significant measure at the present time of assessment than actual achievement. This is meant in the sense that the primary problem of home-based instruction at the present time is, surely, to find ways to reduce the very large dropout rates encountered in almost all forms of home-based instruction.
Apparently even the gifted kids were concerned about grades and credits, and
even Suppes wasn't sure that his studies had shown significant achievement.
It seems a far cry from demonstrating that somewhat less than half of kids at
around 6th-7th grade age level with stratospheric IQs learned the concepts of
logic and modern algebra, most of those rather slowly and with considerable
prodding and assistance, when in a project directed by a noted "subject
matter expert" like Suppes who had been researching his pedagogy for several
years before he tried this particular experiment, and Herman's claims that
large numbers of ordinary elementary kids (were taught these things by
subject matter experts, but that the teachers didn't understand.
Regular education teachers were never involved in Suppes work, which never
came close to a real classroom because he was looking at computer-based
alternatives.
Furthermore, the "teaching" was done by computers, not by "subject matter
experts", and I can testify personally that the proctors that helped the kids
were hardly subject matter experts, but rather were ordinary grad students
working a side job, not even necessarily with math degrees. I know this,
because having placed the work in the correct decade, I remember personally
visiting the ISSR project at about that time and occasionally over the next
several years (I think I even took one of the lessons, but it's been 30
years). I knew one of Suppes' "proctors", who graduated from the class
before mine at Michigan State, and whose background was in programming more
than in math.
In researching Suppes BTW, I found the following:
http://home4.inet.tele.dk/larsens/psychp.html
which has a bunch of neat ideas on why math teaching fails, though the author
was focusing on computer based education.
Among interesting quotes was the following, which supports my claim that much
of the problem with teaching math to younger and less able kids comes from
the language of abstraction. It agrees well with what I've said (much
disputed by Alberto and to a lesser extent Herman) that teaching word
problems of the sort described is the real stumbling block in making math
work for kids. Adults with mathematical knowledge don't even need the
diagram, but kids didn't understand the problem even with the diagram because
it was too abstract.
>This is in accordance with observations by Cofman (1984), who presented this [graphic omitted] figure to thirteen-year-old low attaining pupils and asked them the following question:
>
>"The diagram shows a small square inside a larger square. The region between the two squares is shaded. The side of the longer square is 1O cms, and the side of the small square is 2 cms. Calculate the perimeter of the shaded region."
>
>The pupils knew beforehand how to calculate the perimeter of a square, but nevertheless the majority were not able to solve the problem. Then the question was rephrased:
>
>"If you were asked to fence in the shaded part in the diagram, what would be the total length of the fence needed?"
>
>Now all the pupils answered correctly. It is reasonable to hypothesize that this change in performance was due to the concreteness in the second formulation of the problem. This second formulation turned the mathematical problem into a more real-life situation. Thus the performance of the children might in this case have been supported by inner visualization of the problem; a mental "pictorial" representation so to say.
I hope people find all this useful.
lojbab
Bob LeChevalier
>>specific instructions and not be mere exploration) also seem to support the
>>ordinal concept as well, wherein you sequence the rods in terms of length,
>>and observe that each successive rod differs from the prior one by a unit
>>length equal to the shortest rod. This can be generalized also by building
>>sequences where the length differs by 2 units, etc.
>
>counting, there's no need for any further concept: the rest is, what
>do you know, drill and experience. Our own fingers are well superior
>to rods, and if I had to pick one prop to extend my students' fingers,
>an abacus would be it.
But that fits only the cardinal approach. Your fingers are all different in
a kind of random sort of way, and hence it already takes a bit of abstraction
to think of them as units (albeit an abstraction that most kids make fairly
easily in learning to count, I don't think you could explain Peano's
postulates to a kid using fingers - adding one more after you run out of
fingers doesn't work).
An abacus has units all alike, but you cannot do much to get across the
ordering of those units without first teaching counting. Cuisinaire rods try
to teach the concept without teaching the number names or the counting first,
but instead using "like" and "different", "bigger/smaller" and then working
into number using ordinal, cardinal and measurement, ALL of which takes place
before you assign a single number name (or rather, the number names are the
names of the colors of the rods, since each rod length is a particular
color).
lojbab
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
.................
>>I was at a university where there was a summer institute
>>for high school teachers of mathematics. These were not
>>the strongest mathematically among high school teachers,
>>but also not at the weak end. One of those involved in
>>teaching it estimated the proportion who could manage to
>>learn, under any conditions, the concepts in the basic
>>rigorous undergraduate courses, what are the foundations
>>of analysis and algebra, was only about 10%.
>Why would someone expect that most people could learn a couple of years worth
>of undergraduate math in a summer institute?
Read the sentence more carefully. Notice the words, "under
any conditions"; these were the words which this professor
who taught in the summer session used.
Methinks it is necessary to
>figure out which of those concepts are truly IMPORTANT in order to be able to
>teach what teachers actually are being called upon to do in the curriculum,
The concepts are the important part. Learning the details
and manipulations does not help learn the concepts, but the
reverse is true, and also make it possible to derive the
manipulations quickly.
>and then to hide the remainder under a "you probably don't need to understand
>why, though it would be nice, but teach it THIS way".
You cannot be told HOW to teach concepts. The best you can
be trained to do is to present the textbook material verbatim,
but this will not greatly help the student who cannot get it
without any assistance from the instructor. Nor can such a
person do a reasonable job of grading the course.
.............
>>Didn't anyone tell them that arithmetic is not mathematics,
>>and not at all basic in mathematics?
>I'm sure they did, but just as you tell us, we don't believe you. We think
>that we learned math in school, in my case I studied math in college, and
>arithmetic was basic to almost everything I did in my college math classes
>taught by the math department.
Way back when I first started teaching, and there was still
no dumbing down of the college preparatory math courses,
we all realized that the ONLY real mathematics preparation
the students had was the old "Euclid" geometry course.
This had essentially no computation.
Perhaps not basic in the sense that you mean,
>but doing a math problem and even many kinds of proofs without knowing
>arithmetic is pretty near impossible.
>The real problem is that people learn math to solve problems, and you (and
>Alberto) consider math to be a thing unto itself,
Mathematics is, but I have also worked with, and taught,
many people for whom mathematics is a "service" subject.
What they need is to take their real problems and formulate
them in mathematical terms. There is no point in teaching
people how to solve special cases which are unlikely to
occur, but there is much point in teaching people who are
not mathematically strong to use mathematics to state their
problems.
I have stated that I would make the mathematics requirement
for college admission the ability to take moderately long
(a paragraph or two) word problems and translate them into
symbols, so that humans and machines can solve the formulated
problem. In other words, they need to be able to speak and
write declarative sentences and paragraphs in that language.
Bob LeChevalier
>In article <ilh6iu0gv0ivkujng...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>I was at a university where there was a summer institute
>>>for high school teachers of mathematics. These were not
>>>the strongest mathematically among high school teachers,
>>>but also not at the weak end. One of those involved in
>>>teaching it estimated the proportion who could manage to
>>>learn, under any conditions, the concepts in the basic
>>>rigorous undergraduate courses, what are the foundations
>>>of analysis and algebra, was only about 10%.
>
>>Why would someone expect that most people could learn a couple of years worth
>>of undergraduate math in a summer institute?
>
>Read the sentence more carefully. Notice the words, "under
>any conditions"; these were the words which this professor
>who taught in the summer session used.
Then I think that professor was talking through his hat. Teachers going to
school to take a summer extension class are not especially motivated to try
to learn new subject matter in the first place - they want to learn how to
teach better, and they don't expect to work beyond the limits of the end of
the class. No one could have any idea what they would do if they were full
time students studying the subject matter for its own sake.
> Methinks it is necessary to
>>figure out which of those concepts are truly IMPORTANT in order to be able to
>>teach what teachers actually are being called upon to do in the curriculum,
>
>The concepts are the important part. Learning the details
>and manipulations does not help learn the concepts, but the
>reverse is true, and also make it possible to derive the
>manipulations quickly.
But the teachers don't much care about the concepts in an abstract sense, or
if they do, it is ONLY insofar as it is related to their JOB which is to
teach kids. My long post with many cites today showed that this was a
problem with the new math training seminars as well. Professors tried to
teach the teachers the math, while the teachers were trying to learn how to
teach the math. Absent any pragmatic application to their needs, they are
unlikely to evidence understanding.
>>and then to hide the remainder under a "you probably don't need to understand
>>why, though it would be nice, but teach it THIS way".
>
>You cannot be told HOW to teach concepts.
But that is what the teachers came to the seminar to learn. Their JOB is to
teach the concepts. If you can't teach them how, then you are wasting their
time. Even if they learn the concepts themselves, your attitude tells them
that they are wasting their time, so why bother to learn what you are
teaching them?
>The best you can
>be trained to do is to present the textbook material verbatim,
They already know how to do that, and don't need your seminars. Why would
someone be surprised if only 10% showed the ability to learn under conditions
so hostile to their interests?
lojbab
Bob LeChevalier
>>The idea that addition and multiplication were commutative and even
>>the distributive property already simply was obvious to them before
>>they ever learned the names for such things.
>
>Lots of things are obvious which are not so. One of the
>important points in mathematics is that it is necessary to
>at least for someone to be able to prove something, not
>just to have it obvious.
But kids don't buy that claim, and indeed, until a certain age, they have no
trouble considering absolute contradictions to be true (Santa Claus lives at
the North Pole. I saw Santa Claus at the mall. Daddy is Santa Claus).
Many adults as well believe their hunches first, which is why the "birthday
problem"is a good lesson, as I mentioned in another post. But it is not a
lesson kids will understand; they have no trouble accepting that it is likely
that two kids in their class would have the same birthday.
>I do not know how much, if any, statistics you know, but
>the very important Neyman-Pearson Lemma is conceptually
>accessible to those with a high school algebra level, but
>this understanding is rarely taught at any level, including
>the highest. It is proved, usually in a special case, in
>most mathematical statistics courses, with no insight
>whatever into why it holds.
Looking it up just now, I think that it is not conceptually accessible
primarily because kids have no concept of probability that would have it be
meaningful. To high school kids, probability is about coin flipping, dice
rolling, or maybe if they are ambitious, figure poker or bridge hand odds.
I first saw the kinds of terminology used in the discussions of the lemma in
my senior level calculus-base probability course, and I have to admit that I
understood NOTHING in the class - the "D" I got was a pure gift. I then took
the non-calculus based service course in probability and statistics, and I
pretty much understood it all at the time (though I've forgotten most of it),
and I marveled at how they taught absolutely NOTHING of the stuff in the
other course, The disconnect between the two versions of what the subject of
"probability and statistics" was stark, considering that the two courses were
taught by the same professors.
The lemma is something that MIGHT have been conceptually available to me,
having taken college level math classes, but not as a high school student. I
simply did not view mathematics in a way that allowed me to look at such
formulations as meaningful in any way. And it was not for lack of trying,
since I was the lone kid in my high school who was set to work on that kind
of material (I read Polya, and an introduction to Topology, some number
theory, and a few other books in high school, and none of it really made
sense to me. Even though I had taken SMSG geometry and knew what proofs were
about, I saw them ONLY as applicable to geometry. Algebra was for solving
problems.
I was the top math student in our high school by far, but that stuff was way
out of my league then.
>>I don't think though that naming properties is what Herman thinks of
>>as the *new math.*
>
>You are quite right. However, the properties are important
>and they should be known, together with examples of situations
>where they do not always hold. I cannot think of any easier
>words to describe them.
Actually, we learned that they did not always hold in high school algebra.
But only by being presented with examples, which did not really sink in. It
was not until matrices (which I did not get until college) that I saw
meaningful-to-me cases where they did not hold, and I remember it being a
stunning inside when I realized it.
> I wish, in a way, that I could get hold of
>>something that showed exactly what the very first class contained
>>and who taught it and how he did so. I don't think that what has
>>been kept or even what was tried in *most* schools was what the
>>program started out as.
>
>It might very well have been kept somewhere. I am not
>familiar with education literature.
>
>You could look at the serious of books produced by Suppes
>in the late 50s or early 60s to find something intended
>from first grade on. It was intended to be easier to teach,
>but without losing the essence.
From Suppes' autobiography:
http://www.stanford.edu/~psuppes/autobio1.html
(the quoted text is from page 14, 19 and following pages)
>In 1956 my oldest child, Patricia, entered kindergarten and my interests in applications were once again stimulated, in this ease to thinking about the initial learning of mathematical concepts by children. In collaboration with Newton Hawley, who was and still is a member of the mathematics faculty at Stanford, we began the following year, when our daughters were both in the first grade, the informal introduction of constructive geometry. At that time very little geometry was taught in the primary grades. A brief description of this first effort is to be found in Hawley and Suppes (1959g), but, more importantly, we went on to write two textbooks for primary-grade students in geometry, which have since been translated into French and Spanish (1960c, 1960d).
>Education and Computers
>
>In the section on mathematical concept formation in children I mentioned the beginning of my interests in education in 1956 when my oldest child, Patricia, entered kindergarten. I cited there the work in primary-school geometry. An effort, also noted but briefly, that was much more sustained on my part was work in the basic elementary-school mathematics curriculum. This occupied a fair portion of my time between about 1956 and the middle of the sixties and led to publication of a basic elementary-school mathematics textbook series, Sets and Numbers, which was one of the more radical of the ‘new math’ efforts. Unlike many of my colleagues in mathematics and science who became interested in school curriculum after Sputnik, I had a genuine interest in the psychological and empirical aspects of learning and a traditional interest in knowing what had been done before.
...
>I have already mentioned the important influence of Estes during the year 1955-1956 at the Center for Advanced Study in the Behavioral Sciences. The continuation of the work in learning with applications to multiperson interactions and to mathematical concept formation in children was intellectually a major part of my life during the ten years ending in 1965. The work with Estes continued; we spent many summers together. We planned a monograph as the outgrowth of our work but for various reasons did not complete it. We did write the two long technical reports that were eventually published in shortened form as papers. But the extent of Estes’s influence on my thinking during this period is underestimated by referring simply to the publication of two papers.
Another page among Suppes' site at Stanford lists his early papers on New
Math. Someone with access to a suitable library may be able to find them
>1964) The formation of mathematical concepts in primary-grade children. In A. H. Passow & R. R. Leeper (Eds.), Papers from the ASCD Eighth Curriculum Research Institute, 99-119.
>
>(1964) Modern learning theory and the elementary-school curriculum. American Educational Research Journal, 1, 79-93. Reprinted in H. C. Lindgren (Ed.), Readings in Educational Psychology. New York: Wiley, 207-222. Reprinted also in R. Ripple (Ed.), Readings in Learning and Human Abilities. New York: Harper & Row, 1971. Reprinted also in H. C. Lindgren & F. Lindgren (Eds.), Current Readings in Educational Psychology, 2nd edition. New York: Wiley, 1971, 216-230. Reprinted also in the Bobbs-Merrill Reprint Series in Psychology, P-810, Prod. No. 69065. Japanese translation in W. H. Holtzman (Ed.), Computer-assisted Instruction, Testing, and Guidance. New York: Harper & Row, 1970.
>
>(1964) The ability of elementary-school children to learn the new mathematics. Theory into Practice, 3, 57-61.
>
>(1963) Set theory in the primary grades. New York State Mathematics Teacher's Journal, 13, 46-53. Reprinted in J. J. Gallagher (Ed.), Teaching Gifted Students: A book of readings. Boston: Allyn & Bacon, 1965.
>
>
>(1962) Mathematical logic for the schools. The Arithmetic Teacher, 9, 396-399. Reprinted in J. J. Gallagher (Ed.), Teaching Gifted Students: A book of readings. Boston: Allyn & Bacon, 1965.
>
>With S. Hill. (1962) The concept of set. The Grade Teacher, 79, 5l, 86-90.
>
>With B. McKnight. (1961) Sets and numbers in grade one, 1959-1960. The Arithmetic Teacher, 8, 287-290.
>
>With N. Hawley. (1960) Geometry for Primary Grades. Book 2. San Francisco: Holden-Day, 126 pp. Spanish translation: Geometr’a para los Grados Primarios. Libro 2. San Juan, Puerto Rico: Editorial Departamento de Instrucci—n Pœblica, 1966, 126 pp.
>
>With N. Hawley. (1960) Geometry for Primary Grades. Book 1. San Francisco: Holden-Day, 127 pp. Spanish translation: Geometr’a para los Grados Primarios. Libro 1. San Juan, Puerto Rico: Editorial Departamento de Instrucci—n Pœblica, 1964. 126 pp. French translation: Géométrie pour Classes Elémentaires. Livre 1. Montreal, Canada: Gontran Trottier, 1965.
>
>With N. Hawley. (1959) Geometry in the first grade. American Mathematical Monthly, 66, 505-506.
His first CAI work in the 60s was in teaching younger kids arithmetic, using
lots of drill and a behaviorist approach, which sounds exactly the opposite
of what you've described, whereas his research in the 70s with the very
gifted, does match your description. I've just now found that he had done
some earlier work in the 60s with elementary kids, described in the paper at
http://www.bambi.net/bob/suppes.pdf
but this again was 40 kids of 2nd grade age level selected specifically for
their higher IQs 122-166, and they excluded 18 kids in the 105-126 IQ range.
With these ultrabright kids he did report being able to cover 1 3/4 years of
subject matter in 6 months (which seems unexceptional for kids that bright),
but as I said, his methodology sounds exactly the opposite of what you would
favor, though it fits better with Alberto. The kids were doing over 100
problems per day to achieve that pacing.
Bob Davis at Syracuse did the Madison Project in the early 60s dealing with
3rd through 8th graders, but I have been unable to find anything on the
nature of the mathematics that was taught other than the brief description of
his book:
http://www.enc.org/resources/records/full/0,1240,002529,00.shtm
I haven't found anything earlier than that where new math researchers were
working below the high school level.
lojbab
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>Finally, some useful information, to which I add more below.
>toto <scar...@wicked.witch> wrote:
>>On Wed, 03 Jul 2002 03:49:24 GMT, Bob LeChevalier
>><loj...@lojban.org> wrote:
....................
>The textbooks were not practical for the classroom (I know because I studied
>from the SMSG Geometry text, and also tutored other kids, some quite bright,
>using it).
Which SMSG geometry text? One of the SMSG panelists told me
that the group assigned to produce this produced a version
of "Euclid", which made explicit all the assumptions which
Euclid did not realize were necessary, but were pointed out
by Hilbert in the 19th century. The old Euclid books were
the standard texts before WWII.
Those in charge on the government side essentially rejected
this, telling them that it would be included in the list of
texts, but that they should produce another one, with few
proofs, and connecting geometry with algebra and computation.
This is the more used one.
The old Euclid books were used in the standard college
preparatory program before. To some extent, they were
even teacher proof, as theorems and proofs were carefully
explained, but naturally there was nothing teaching
students how to go about proving theorems. I use the term
"naturally" because this is something we do not know how
to do, and probably never will. Creativity can be evoked,
but not taught.
................
>>Tom Lehrer's 1965 song New Math, lam-
>>pooning the pretentious language used to justify an inability to
>>calculate, had the mathematical community itself laughing at the
>>follies committed in the name of promoting a better understanding
>>of mathematics.
>Even mathematicians apparently felt that the ability to calculate was still
>important.
Some of those in such fields as classical analysis, where
the foundations do not intrude that much, seem to think
so. Some of them seem unable to recognize that there are
problems. But Cantor, Peano, Dedekind, and Hilbert came
from classical analysis, and had a great deal to do with
setting up the present foundations. Landau's book on the
logical development of the number systems is called
_Foundations of Analysis_. The first part, on the positive
integers, should be required for all. I believe that, with
small modifications, it can be used to introduce an
understanding of arithmetic.
There are lots of first-rate mathematicians who are very
poor at arithmetic. As I have stated, I am not one of
those. That mathematicians tend to be good at calculation
is, to some extent, due to proficiency at calculation often
being the criterion for recommendation to go into mathematics.
If the one who can understand addition and multiplication but
is slow at it is discouraged, or even denied, strong solid
mathematics, this will continue.
>> On the other hand, SMSG and its imitators were also guilty
>>of some pointless pedantry, ridiculous even if logically correct:
>>"Write the numeral that names the number solving 3x -7 = 8," for
>>example.
>Much of Herman's argument seems like just such pointless pedantry - it still
>remains unclear what he WOULD have teachers teach, and how.
I probably would not state it that way, but it is extremely
important to realize that numbers are not strings of characters.
The use of different bases is of this nature; a number is the
same number no matter how it is represented.
As education should be for the future, it should be made
clear very early that these distinctions are needed. It
is very hard to unlearn later.
>>Had SMSG really been tried? The mass of American tea-
>>chers -- and children -- were not in the end exposed to, let alone
>>taught, what the SMSG mathematicians prescribed.
>Teachers cannot have failed to learn what they were never exposed to.
When the new math was introduced, nobody expected the
teachers to know it. What was expected was that teachers
could fairly quickly learn what elementary school children
can manage. Nobody thought that they would have problems,
and if they had any understanding of mathematical concepts,
they would not.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <ilh6iu0gv0ivkujng...@4ax.com>,
>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>>I was at a university where there was a summer institute
>>>>for high school teachers of mathematics. These were not
>>>>the strongest mathematically among high school teachers,
>>>>but also not at the weak end. One of those involved in
>>>>teaching it estimated the proportion who could manage to
>>>>learn, under any conditions, the concepts in the basic
>>>>rigorous undergraduate courses, what are the foundations
>>>>of analysis and algebra, was only about 10%.
>>>Why would someone expect that most people could learn a couple of years worth
>>>of undergraduate math in a summer institute?
>>Read the sentence more carefully. Notice the words, "under
>>any conditions"; these were the words which this professor
>>who taught in the summer session used.
>Then I think that professor was talking through his hat. Teachers going to
>school to take a summer extension class are not especially motivated to try
>to learn new subject matter in the first place - they want to learn how to
>teach better, and they don't expect to work beyond the limits of the end of
>the class. No one could have any idea what they would do if they were full
>time students studying the subject matter for its own sake.
In this case, you are completely wrong. The summer institute
was not for the purpose of "improving teaching", but for the
purpose of increasing mathematical understanding.
>> Methinks it is necessary to
>>>figure out which of those concepts are truly IMPORTANT in order to be able to
>>>teach what teachers actually are being called upon to do in the curriculum,
>>The concepts are the important part. Learning the details
>>and manipulations does not help learn the concepts, but the
>>reverse is true, and also make it possible to derive the
>>manipulations quickly.
>But the teachers don't much care about the concepts in an abstract sense, or
>if they do, it is ONLY insofar as it is related to their JOB which is to
>teach kids.
Those with that attitude will not be able to learn any
concepts. It might be possible to completely reeducate
them mathematically; I do not believe it has been tried.
A concept is not a set of words, to be memorized and
regurgitated when requested. It is not a set of facts.
It is not composed of procedures. If we want students to
learn the concepts, even later, it is at least highly
desirable that those who teach memorization and routine
not be allowed to teach the students.
My long post with many cites today showed that this was a
>problem with the new math training seminars as well. Professors tried to
>teach the teachers the math, while the teachers were trying to learn how to
>teach the math. Absent any pragmatic application to their needs, they are
>unlikely to evidence understanding.
Professors were quite clear that they were teaching the
math. It is the educationists who believe that there
is a "way to teach" which does not involve really knowing
the subject who are the cause of the problem. Those who
understand the concepts can teach courses for which they
do not have prior knowledge of the details, but those who
only know the details may be in worse shape than if they
knew nothing.
>>>and then to hide the remainder under a "you probably don't need to understand
>>>why, though it would be nice, but teach it THIS way".
>>You cannot be told HOW to teach concepts.
>But that is what the teachers came to the seminar to learn. Their JOB is to
>teach the concepts.
No, they came to learn the concepts, and they were told
this in advance. Those who do not know the concepts at
best can have the students study, but they cannot teach.
If you can't teach them how, then you are wasting their
>time. Even if they learn the concepts themselves, your attitude tells them
>that they are wasting their time, so why bother to learn what you are
>teaching them?
If they learn the concepts, they will understand why these
concepts give insight, and they have a good chance of being
able to teach them.
>>The best you can
>>be trained to do is to present the textbook material verbatim,
>They already know how to do that, and don't need your seminars. Why would
>someone be surprised if only 10% showed the ability to learn under conditions
>so hostile to their interests?
It is not the conditions, but what. If they cannot learn the
concepts, or at least recognize that they do not know them,
but still will try to get the children to learn them, they do
more harm than good. There were many coaches and others who
taught Euclid from the old books, and at least they made no
attempt to reduce the students to their level.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>The idea that addition and multiplication were commutative and even
>>>the distributive property already simply was obvious to them before
>>>they ever learned the names for such things.
>>Lots of things are obvious which are not so. One of the
>>important points in mathematics is that it is necessary to
>>at least for someone to be able to prove something, not
>>just to have it obvious.
>But kids don't buy that claim, and indeed, until a certain age, they have no
>trouble considering absolute contradictions to be true (Santa Claus lives at
>the North Pole. I saw Santa Claus at the mall. Daddy is Santa Claus).
If you present it honestly, they will.
>Many adults as well believe their hunches first, which is why the "birthday
>problem"is a good lesson, as I mentioned in another post. But it is not a
>lesson kids will understand; they have no trouble accepting that it is likely
>that two kids in their class would have the same birthday.
>>I do not know how much, if any, statistics you know, but
>>the very important Neyman-Pearson Lemma is conceptually
>>accessible to those with a high school algebra level, but
>>this understanding is rarely taught at any level, including
>>the highest. It is proved, usually in a special case, in
>>most mathematical statistics courses, with no insight
>>whatever into why it holds.
>Looking it up just now, I think that it is not conceptually accessible
>primarily because kids have no concept of probability that would have it be
>meaningful. To high school kids, probability is about coin flipping, dice
>rolling, or maybe if they are ambitious, figure poker or bridge hand odds.
It can be "made obvious" at the level of coin flipping or
dice rolling, testing whether the coin or die is "honest".
>I first saw the kinds of terminology used in the discussions of the lemma in
>my senior level calculus-base probability course, and I have to admit that I
>understood NOTHING in the class - the "D" I got was a pure gift. I then took
>the non-calculus based service course in probability and statistics, and I
>pretty much understood it all at the time (though I've forgotten most of it),
>and I marveled at how they taught absolutely NOTHING of the stuff in the
>other course, The disconnect between the two versions of what the subject of
>"probability and statistics" was stark, considering that the two courses were
>taught by the same professors.
If someone taught a methods course has a statistics problem,
it will be very hard to get at the problem. The concepts
are not taught by doing lots of calculations. Probably both
of your courses were computational without understanding.
>The lemma is something that MIGHT have been conceptually available to me,
>having taken college level math classes, but not as a high school student.
See if you can figure it out with the above statement of how
low-level it can be presented. But you will need to use
randomized procedures to explain it.
I
>simply did not view mathematics in a way that allowed me to look at such
>formulations as meaningful in any way. And it was not for lack of trying,
>since I was the lone kid in my high school who was set to work on that kind
>of material (I read Polya, and an introduction to Topology, some number
>theory, and a few other books in high school, and none of it really made
>sense to me. Even though I had taken SMSG geometry and knew what proofs were
>about, I saw them ONLY as applicable to geometry. Algebra was for solving
>problems.
Did they not present induction in college algebra?
But proofs belong in first grade arithmetic. Then they
become part of the background, and will get used elsewhere.
If not that early, certainly by middle school.
>I was the top math student in our high school by far, but that stuff was way
>out of my league then.
>>>I don't think though that naming properties is what Herman thinks of
>>>as the *new math.*
>>You are quite right. However, the properties are important
>>and they should be known, together with examples of situations
>>where they do not always hold. I cannot think of any easier
>>words to describe them.
>Actually, we learned that they did not always hold in high school algebra.
>But only by being presented with examples, which did not really sink in.
How else does one learn that something does not always hold?
It
>was not until matrices (which I did not get until college) that I saw
>meaningful-to-me cases where they did not hold, and I remember it being a
>stunning inside when I realized it.
A good example to show the lack of commutativity is
putting on a sock and putting on a shoe on the same foot.
On different feet, they commute. It is not hard to give
kindergarten level examples. The one who understands the
concepts sees this type of situation everywhere; algebra
is not just for numbers, and should not be taught that way.
General concepts are easier.
Shana
After reading some of your postings about higher mastery, I think some
are getting too wrapped up in the idea that higher mastery is directly
linked to a student's ability to execute a certain concept. I think
the article was trying to go deeper than that when refering to higher
mastery of concepts. It is my understanding that when a student is
able to internalize a concept, perform a concept, share a concept with
others, and explain their thinking in relevant terms to their
understanding of the concept and not the teacher's understanding, then
the child has acheived a higher mastery of that concept.
Bob LeChevalier
>In article <4cp7iu4jcbdorm0kn...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>Then I think that professor was talking through his hat. Teachers going to
>>school to take a summer extension class are not especially motivated to try
>>to learn new subject matter in the first place - they want to learn how to
>>teach better, and they don't expect to work beyond the limits of the end of
>>the class. No one could have any idea what they would do if they were full
>>time students studying the subject matter for its own sake.
>
>In this case, you are completely wrong. The summer institute
>was not for the purpose of "improving teaching", but for the
>purpose of increasing mathematical understanding.
That may have been the professor's purpose, but what was the teachers'
purpose in taking the institute? If it was not to find ways to improve their
teaching, I'd be VERY surprised.
>>But the teachers don't much care about the concepts in an abstract sense, or
>>if they do, it is ONLY insofar as it is related to their JOB which is to
>>teach kids.
>
>Those with that attitude will not be able to learn any
>concepts.
Almost NOBODY learns things just because they are there. They learn things
for a reason, and you need to adapt your teaching to fit their reason for
learning, OR give them motivation to learn what you wish to teach. If you
fail at the latter, it is YOUR failure and not theirs. After all, they are
paying you to give them the education THEY are seeking.
>A concept is not a set of words, to be memorized and
>regurgitated when requested. It is not a set of facts.
>It is not composed of procedures.
Irrelevant.
>If we want students to
>learn the concepts, even later, it is at least highly
>desirable that those who teach memorization and routine
>not be allowed to teach the students.
Fine. Then you have to show them another way to teach it, bearing in mind
that their students will need to learn what you teach them. Your attitude as
displayed on this forum is enough to pretty well dissuade most teachers from
bothering to try to learn from you - after all, you assume from the start
that they can't learn the material, and assume that they want only to teach
memorization and routine. Alberto takes the opposite approach; he may work
them to death doing so, but he will make all personal effort to make SURE
that any student making the effort will learn what needs to be learned.
> My long post with many cites today showed that this was a
>>problem with the new math training seminars as well. Professors tried to
>>teach the teachers the math, while the teachers were trying to learn how to
>>teach the math. Absent any pragmatic application to their needs, they are
>>unlikely to evidence understanding.
>
>Professors were quite clear that they were teaching the
>math. It is the educationists who believe that there
>is a "way to teach" which does not involve really knowing
>the subject who are the cause of the problem.
Knowing the subject is not sufficient though. Oversimplifying, I imagine
that the professors taught a basic concept and moved on to the next. The
teachers learned and understood the first concept and tried to figure out how
they could teach it. Until they learned how to use what they had learned for
THEIR purposes, they were not ready to learn more towards the teachers'
purposes. This is probably the major difference between teaching kids and
teaching adults. Adults are in there with a goal that THEY have selected,
and they are paying you to help them reach that goal. Kids rarely have clear
goals (other than to get a good grade), and it is either their parents or the
state that is paying, so they do not consider you accountable to them.
>>>You cannot be told HOW to teach concepts.
>
>>But that is what the teachers came to the seminar to learn. Their JOB is to
>>teach the concepts.
>
>No, they came to learn the concepts, and they were told
>this in advance.
Tell them all you want. If you are not giving them something they are
finding useful, then like all students, they will tune you out.
> If you can't teach them how, then you are wasting their
>>time. Even if they learn the concepts themselves, your attitude tells them
>>that they are wasting their time, so why bother to learn what you are
>>teaching them?
>
>If they learn the concepts, they will understand why these
>concepts give insight, and they have a good chance of being
>able to teach them.
Per the quotes I cited, even when the teachers grasped the concepts, they
were useless to them because no one had figured out what to do with them in
the classroom. Apparently Suppes made some effort, but it remains unclear
that he explained in a way useful to others just what he figured out, and
whether it applies only to the gifted or to all the kids. Someone would have
to read his papers to find out, and I don't have access.
>>>The best you can
>>>be trained to do is to present the textbook material verbatim,
>
>>They already know how to do that, and don't need your seminars. Why would
>>someone be surprised if only 10% showed the ability to learn under conditions
>>so hostile to their interests?
>
>It is not the conditions, but what. If they cannot learn the
>concepts, or at least recognize that they do not know them,
>but still will try to get the children to learn them, they do
>more harm than good.
Their job is to teach the material directed by the state, whether that
teaching does harm or good. If there is a way that the can teach it that
does not harm, they want to learn that way. The concepts you are teaching
them MUST be presented as a step towards that learning AND the professor
needs to make it clear that he understand why they are trying to learn.
> There were many coaches and others who
>taught Euclid from the old books, and at least they made no
>attempt to reduce the students to their level.
One of the papers I read said explicitly that long before it was eliminated,
Euclid's geometry was being dumbed down to worthlessness.
lojbab
Bob LeChevalier
>In article <c2q7iuoosjdbgr9s3...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>I first saw the kinds of terminology used in the discussions of the lemma in
>>my senior level calculus-base probability course, and I have to admit that I
>>understood NOTHING in the class - the "D" I got was a pure gift. I then took
>>the non-calculus based service course in probability and statistics, and I
>>pretty much understood it all at the time (though I've forgotten most of it),
>>and I marveled at how they taught absolutely NOTHING of the stuff in the
>>other course, The disconnect between the two versions of what the subject of
>>"probability and statistics" was stark, considering that the two courses were
>>taught by the same professors.
>
>If someone taught a methods course has a statistics problem,
>it will be very hard to get at the problem. The concepts
>are not taught by doing lots of calculations. Probably both
>of your courses were computational without understanding.
Presuming that you know the textbooks, the higher level course used Dwass's
_Probability: Theory and Application_ The class was all theory and we didn't
compute much of anything; the homework was theory plus one specifically
chosen application problem each day. Alas, I seldom saw any connection
between the theory and the application, and I don't pretend that I understood
the concepts. At 1/2 the speed, and having been prepared in advance to take
a mathematical theory class aimed at majors (which I had never taken before
then, so I was over my head before I knew it, and gave up in order to spend
time on my classes required for my major), I might have done better. But I
came to college incapable of learning from a lecture, and I hit that class in
my junior year (which was my second year at Michigan State) never having
acquired the skill of getting useful information from a lecture (half the
time I dozed off).
>>The lemma is something that MIGHT have been conceptually available to me,
>>having taken college level math classes, but not as a high school student.
>
>See if you can figure it out with the above statement of how
>low-level it can be presented. But you will need to use
>randomized procedures to explain it.
Too far back for me, and I don't have time to spend on figuring it out now.
I vaguely see that it pertains to the application you suggested, which might
even interest the kids if one started with a loaded die and demonstrated the
point of determining that it was.
> I
>>simply did not view mathematics in a way that allowed me to look at such
>>formulations as meaningful in any way. And it was not for lack of trying,
>>since I was the lone kid in my high school who was set to work on that kind
>>of material (I read Polya, and an introduction to Topology, some number
>>theory, and a few other books in high school, and none of it really made
>>sense to me. Even though I had taken SMSG geometry and knew what proofs were
>>about, I saw them ONLY as applicable to geometry. Algebra was for solving
>>problems.
>
>Did they not present induction in college algebra?
I didn't take college algebra. They discussed induction in high school
algebra, but I had no interest in proof, and never figured out why I should
care that they used induction to prove this or that. I understood what they
had done, but only as a demonstration; I could prove things like trig
identities that were straightforward manipulation with cute substitution
tricks, but I don't think I even tried any problems that asked for a proof by
induction - how should I know how to do it?
I never, in math department courses, or any other math courses, figured out
why I needed to know how the mathematicians had proven it, rather than merely
THAT they had proven it, and I had no curiosity for the question. It was
proven, so how could I use it, was the main question I had. NO ONE in my
university years explained the point, which I perhaps vaguely started to
grasp around 15 years later reading books like Goedel Escher Bach or The
Mathematical Experience. I got a little more handle on it around 12 years
ago when I ran into someone who was trying to devise a mathematics based on
denying the Axiom of Equality, and was able to explain to me in layman's
terms why one would do such a thing.
>But proofs belong in first grade arithmetic. Then they
>become part of the background, and will get used elsewhere.
In my experience, there is no "background" to a first grader. Everything is
foreground, and you would only confuse him.
>>>>I don't think though that naming properties is what Herman thinks of
>>>>as the *new math.*
>
>>>You are quite right. However, the properties are important
>>>and they should be known, together with examples of situations
>>>where they do not always hold. I cannot think of any easier
>>>words to describe them.
>
>>Actually, we learned that they did not always hold in high school algebra.
>>But only by being presented with examples, which did not really sink in.
>
>How else does one learn that something does not always hold?
Since it wasn't made clear that we should CARE whether they hold or not, it
was noise. Students often learn what they are told merely because they are
told to. Figuring out why requires them to believe that the reason why is
accessible to them and important to them. To most kids, the only such reason
is "because the teacher says so, and you'll flunk if you don't".
>The one who understands the
>concepts sees this type of situation everywhere; algebra
>is not just for numbers, and should not be taught that way.
>General concepts are easier.
I first saw that (algebra is not just for numbers) sometime around 1985, 11
years after finishing college.
lojbab
Bob LeChevalier
>In article <n8o6iu05m91drl0e2...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>Finally, some useful information, to which I add more below.
>
>>toto <scar...@wicked.witch> wrote:
>>>On Wed, 03 Jul 2002 03:49:24 GMT, Bob LeChevalier
>>><loj...@lojban.org> wrote:
> ....................
>
>>The textbooks were not practical for the classroom (I know because I studied
>>from the SMSG Geometry text, and also tutored other kids, some quite bright,
>>using it).
>
>Which SMSG geometry text? One of the SMSG panelists told me
>that the group assigned to produce this produced a version
>of "Euclid", which made explicit all the assumptions which
>Euclid did not realize were necessary, but were pointed out
>by Hilbert in the 19th century. The old Euclid books were
>the standard texts before WWII.
>
>Those in charge on the government side essentially rejected
>this, telling them that it would be included in the list of
>texts, but that they should produce another one, with few
>proofs, and connecting geometry with algebra and computation.
>This is the more used one.
I don't recall that the book I used mentioned Euclid at all, but it most
certainly was proofs, proofs and more proofs, seldom with motivation
presented for why one would care to prove whatever it was. The lack of
connection with algebra or anything else that had come before in math totally
baffled kids. Most who made it simply took the attitude that it didn't have
to make sense - they were learning it because that is what was in the
textbook. Only the unit on vectors near the end of the book had some
calculation.
>The old Euclid books were used in the standard college
>preparatory program before. To some extent, they were
>even teacher proof, as theorems and proofs were carefully
>explained, but naturally there was nothing teaching
>students how to go about proving theorems.
Which was the problem with the SMSG book as well.
>I use the term
>"naturally" because this is something we do not know how
>to do, and probably never will. Creativity can be evoked,
>but not taught.
Polya made an effort, and I understood a little of what he said, but it was
too far above my level at the time to really get it.
>When the new math was introduced, nobody expected the
>teachers to know it. What was expected was that teachers
>could fairly quickly learn what elementary school children
>can manage. Nobody thought that they would have problems,
>and if they had any understanding of mathematical concepts,
>they would not.
They expected the teachers to learn things without having had any background.
Other than being able to read somewhat more complex info a little faster, the
teacher will likely take as long to learn as the kids. Why anyone would
expect a teacher, even one just out of college, to learn in a summer course
any more than a college student would learn in a summer course, baffles me.
lojbab
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <4cp7iu4jcbdorm0kn...@4ax.com>,
>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>Then I think that professor was talking through his hat. Teachers going to
>>>school to take a summer extension class are not especially motivated to try
>>>to learn new subject matter in the first place - they want to learn how to
>>>teach better, and they don't expect to work beyond the limits of the end of
>>>the class. No one could have any idea what they would do if they were full
>>>time students studying the subject matter for its own sake.
>>In this case, you are completely wrong. The summer institute
>>was not for the purpose of "improving teaching", but for the
>>purpose of increasing mathematical understanding.
>That may have been the professor's purpose, but what was the teachers'
>purpose in taking the institute? If it was not to find ways to improve their
>teaching, I'd be VERY surprised.
It was to improve their ability to teach. NOTHING was taught
about methods, and this was clear well in advance. They were
there to get a better understanding of the mathematics needed.
This is what is needed in all subjects. Someone who understands
the concepts can decide how to teach, not perform.
>>>But the teachers don't much care about the concepts in an abstract sense, or
>>>if they do, it is ONLY insofar as it is related to their JOB which is to
>>>teach kids.
>>Those with that attitude will not be able to learn any
>>concepts.
>Almost NOBODY learns things just because they are there. They learn things
>for a reason, and you need to adapt your teaching to fit their reason for
>learning, OR give them motivation to learn what you wish to teach. If you
>fail at the latter, it is YOUR failure and not theirs. After all, they are
>paying you to give them the education THEY are seeking.
I question whether the teachers posting here are willing
to take that attitude. But the schools are instilling
that into the children, by teaching how and what, and
expecting that to be regurgitated.
The reason they need to learn the concepts is so they
can understand what is behind what they will be
teaching. I believe that what they were being taught in
those institutes should be taught in high school, but
there has been little move in that direction. However,
a teacher needs to know the basic concepts to understand
the questions.
>>A concept is not a set of words, to be memorized and
>>regurgitated when requested. It is not a set of facts.
>>It is not composed of procedures.
>Irrelevant.
NOT irrelevant. Those drilled in facts and procedures
almost need to start over. Those with strong mathematical
ability can do this in minutes, but the others have major
problems. The old Euclid course was the old introduction,
although there are better ones, and these can come quite
early, and should, but they do not.
>>If we want students to
>>learn the concepts, even later, it is at least highly
>>desirable that those who teach memorization and routine
>>not be allowed to teach the students.
>Fine. Then you have to show them another way to teach it, bearing in mind
>that their students will need to learn what you teach them.
Your attitude as
>displayed on this forum is enough to pretty well dissuade most teachers from
>bothering to try to learn from you - after all, you assume from the start
>that they can't learn the material,
I make no such assumptions. I am willing to put in whatever
effort is required to teach the concepts. What I will not do
is to teach the routine instead. But I cannot MAKE them learn,
nor do I have the right to increase OR DECREASE the stated level
of the course. I do have the right to increase what is taught,
but not decrease it.
and assume that they want only to teach
>memorization and routine.
This is based on years of observation. It shows up quickly,
when they object to being asked to understand, but instead ask
to be given a collection of procedures. It shows up when
prospective high school teachers with two years of calculus
cannot, on a take-home exam, do problems similar to the ones
in previous homework and gone over in class, because they do
not understand derivatives and integrals, but only know the
formulas to calculate them.
This one was told to me; in a class to prepare prospective
high school teachers, not the weakest ones, they were taught
specific items and how to present them, and then taught those
in the remedial classes for college students. One lesson was
on the equation of a line in Euclidean two-space. They were
shown how to derive the formula, which is quite easy, and
were instructed to teach the derivation. On observation, they
taught the formula, which meant the students learned nothing
important. They were told to teach it again, with emphasis on
the why, not the formula.
Now what happens if students are not given this type of course?
It cannot be done for all, as the staff required is too high.
Even if the book has the derivation, they skip over it, and
teach just the formulas. We would do better to have the students
just read the book and ask questions, and not have them teach.
Alberto takes the opposite approach; he may work
>them to death doing so, but he will make all personal effort to make SURE
>that any student making the effort will learn what needs to be learned.
>> My long post with many cites today showed that this was a
>>>problem with the new math training seminars as well. Professors tried to
>>>teach the teachers the math, while the teachers were trying to learn how to
>>>teach the math. Absent any pragmatic application to their needs, they are
>>>unlikely to evidence understanding.
>>Professors were quite clear that they were teaching the
>>math. It is the educationists who believe that there
>>is a "way to teach" which does not involve really knowing
>>the subject who are the cause of the problem.
>Knowing the subject is not sufficient though. Oversimplifying, I imagine
>that the professors taught a basic concept and moved on to the next.
The concept was USED later, quite probably in going on to
the next. One does not teach a concept and just ignore it;
concepts are used.
The
>teachers learned and understood the first concept and tried to figure out how
>they could teach it.
You are assuming that the concept was presented as words to
be memorized. It isn't. You cannot be taught how to teach
a concept; if the presentation and the exercises do not work,
you use the understanding of the concept to look at it from
another angle. And you do not use a time schedule. Having
a daily "lesson plan" to be followed precisely cannot work to
teach concepts.
.................
>>If they learn the concepts, they will understand why these
>>concepts give insight, and they have a good chance of being
>>able to teach them.
>Per the quotes I cited, even when the teachers grasped the concepts, they
>were useless to them because no one had figured out what to do with them in
>the classroom. Apparently Suppes made some effort, but it remains unclear
>that he explained in a way useful to others just what he figured out, and
>whether it applies only to the gifted or to all the kids. Someone would have
>to read his papers to find out, and I don't have access.
>>>>The best you can
>>>>be trained to do is to present the textbook material verbatim,
>>>They already know how to do that, and don't need your seminars. Why would
>>>someone be surprised if only 10% showed the ability to learn under conditions
>>>so hostile to their interests?
>>It is not the conditions, but what. If they cannot learn the
>>concepts, or at least recognize that they do not know them,
>>but still will try to get the children to learn them, they do
>>more harm than good.
>Their job is to teach the material directed by the state, whether that
>teaching does harm or good. If there is a way that the can teach it that
>does not harm, they want to learn that way. The concepts you are teaching
>them MUST be presented as a step towards that learning AND the professor
>needs to make it clear that he understand why they are trying to learn.
This is why we need alternate educational paths. We had students
who understood 50 years ago; we do not have them now.
>> There were many coaches and others who
>>taught Euclid from the old books, and at least they made no
>>attempt to reduce the students to their level.
>One of the papers I read said explicitly that long before it was eliminated,
>Euclid's geometry was being dumbed down to worthlessness.
This only happened because to the educationists' mandate to
teach it to all. I cannot consider an adequate penalty for
someone who lowers the level or content of a course, and this
is exactly what was done by trying to teach it to everyone.
Let the "average people" decide what is a good course, and it
ends up to be junk.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <c2q7iuoosjdbgr9s3...@4ax.com>,
>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>I first saw the kinds of terminology used in the discussions of the lemma in
>>>my senior level calculus-base probability course, and I have to admit that I
>>>understood NOTHING in the class - the "D" I got was a pure gift. I then took
>>>the non-calculus based service course in probability and statistics, and I
>>>pretty much understood it all at the time (though I've forgotten most of it),
>>>and I marveled at how they taught absolutely NOTHING of the stuff in the
>>>other course, The disconnect between the two versions of what the subject of
>>>"probability and statistics" was stark, considering that the two courses were
>>>taught by the same professors.
>>If someone taught a methods course has a statistics problem,
>>it will be very hard to get at the problem. The concepts
>>are not taught by doing lots of calculations. Probably both
>>of your courses were computational without understanding.
>Presuming that you know the textbooks, the higher level course used Dwass's
>_Probability: Theory and Application_
It could still be mainly calculation. Knowing how to calculate
binomial probabilities, etc., which is usually the beginning
part of such a course, is far less important than knowing what
has a binomial distribution, and what might look like it, but
does not.
The class was all theory and we didn't
>compute much of anything; the homework was theory plus one specifically
>chosen application problem each day. Alas, I seldom saw any connection
>between the theory and the application, and I don't pretend that I understood
>the concepts. At 1/2 the speed, and having been prepared in advance to take
>a mathematical theory class aimed at majors (which I had never taken before
>then, so I was over my head before I knew it, and gave up in order to spend
>time on my classes required for my major), I might have done better. But I
>came to college incapable of learning from a lecture, and I hit that class in
>my junior year (which was my second year at Michigan State) never having
>acquired the skill of getting useful information from a lecture (half the
>time I dozed off).
So learn from the textbook, and ask questions. Most faculty
teaching such a course are willing to answer them, or to
provide help otherwise. And, which few students seem to do,
read the section to be discussed BEFORE coming to class.
Professors are not equipped with magic wands which enable the
ideas to penetrate the major barrier to the transmission of
knowledge, 1/4 inch of human skull.
>>>The lemma is something that MIGHT have been conceptually available to me,
>>>having taken college level math classes, but not as a high school student.
>>See if you can figure it out with the above statement of how
>>low-level it can be presented. But you will need to use
>>randomized procedures to explain it.
>Too far back for me, and I don't have time to spend on figuring it out now.
>I vaguely see that it pertains to the application you suggested, which might
>even interest the kids if one started with a loaded die and demonstrated the
>point of determining that it was.
>> I
>>>simply did not view mathematics in a way that allowed me to look at such
>>>formulations as meaningful in any way. And it was not for lack of trying,
>>>since I was the lone kid in my high school who was set to work on that kind
>>>of material (I read Polya, and an introduction to Topology, some number
>>>theory, and a few other books in high school, and none of it really made
>>>sense to me. Even though I had taken SMSG geometry and knew what proofs were
>>>about, I saw them ONLY as applicable to geometry. Algebra was for solving
>>>problems.
>>Did they not present induction in college algebra?
>I didn't take college algebra. They discussed induction in high school
>algebra, but I had no interest in proof, and never figured out why I should
>care that they used induction to prove this or that.
College algebra used to be standard in high school. It
usually was part of the second year of algebra.
So you did not want to know why, and you cut yourself off
from understanding the basic properties of the integers.
Understanding the positive integers as a whole requires
understanding this, and I have suggested that reading a
development of the integers from the Peano postulates be
a mandatory part of the preparation for even first grade
teachers.
Now Euclid did not know this development, but he used a
version of induction. One version, which requires more
than what is needed in a direct axiomatic approach, is that
every non-empty set of integers has a smallest element.
This is what Euclid used.
I understood what they
>had done, but only as a demonstration; I could prove things like trig
>identities that were straightforward manipulation with cute substitution
>tricks, but I don't think I even tried any problems that asked for a proof by
>induction - how should I know how to do it?
It was presented; that was enough reason to consider not
just learning it, but understanding it.
>I never, in math department courses, or any other math courses, figured out
>why I needed to know how the mathematicians had proven it, rather than merely
>THAT they had proven it, and I had no curiosity for the question.
Curiosity may have killed the cat, but not having it is
deadly for education. Young children have it, but it seems
to disappear. Could the way the schools approach things
have anything to do with this?
It was
>proven, so how could I use it, was the main question I had. NO ONE in my
>university years explained the point, which I perhaps vaguely started to
>grasp around 15 years later reading books like Goedel Escher Bach or The
>Mathematical Experience. I got a little more handle on it around 12 years
>ago when I ran into someone who was trying to devise a mathematics based on
>denying the Axiom of Equality, and was able to explain to me in layman's
>terms why one would do such a thing.
>>But proofs belong in first grade arithmetic. Then they
>>become part of the background, and will get used elsewhere.
>In my experience, there is no "background" to a first grader. Everything is
>foreground, and you would only confuse him.
I doubt this, as the first grader can understand structure,
not having been had it drilled into him that one just
memorizes facts and methods and produces these on demand.
It is the one who "knows arithmetic" who is unwilling to
consider learning what the foundations are.
>>>>>I don't think though that naming properties is what Herman thinks of
>>>>>as the *new math.*
>>>>You are quite right. However, the properties are important
>>>>and they should be known, together with examples of situations
>>>>where they do not always hold. I cannot think of any easier
>>>>words to describe them.
>>>Actually, we learned that they did not always hold in high school algebra.
>>>But only by being presented with examples, which did not really sink in.
>>How else does one learn that something does not always hold?
>Since it wasn't made clear that we should CARE whether they hold or not, it
>was noise. Students often learn what they are told merely because they are
>told to. Figuring out why requires them to believe that the reason why is
>accessible to them and important to them.
Learning is for the FUTURE, not the present. Tying it to what
is of present interest does great harm.
To most kids, the only such reason
>is "because the teacher says so, and you'll flunk if you don't".
The schools which teach this should be replaced completely,
or we cannot get educated people.
>>The one who understands the
>>concepts sees this type of situation everywhere; algebra
>>is not just for numbers, and should not be taught that way.
>>General concepts are easier.
>I first saw that (algebra is not just for numbers) sometime around 1985, 11
>years after finishing college.
That is why it belongs in first grade, and not just for
numbers.
Thad Coons
coons
"Alberto Moreira" <junk...@moreira.mv.com> wrote in message
news:4cq5iu8ael4tvla6o...@4ax.com...
>
> Repeating, then: I look at mathematics from a kind of a logicist point
> of view: mathematics is logic. Logic, mind you, is content free. There
> are axioms, there are rules of inference. We apply rules of inference
> to axioms to get theorems. THAT'S IT - there's no "concept" in this,
> only axioms and rules of inference.Mathematics is BUILT, not
> discovered, it has DEFINITIONS, not concept, MODELS, not reality.
> Those definitions FIT the concept which exists OUTSIDE mathematics, in
> the real world we're trying to model with math. The moment you say
> "concept", I say, you're springing out of math into some real-world
> subject, or you're digging in further down into your mathematical
> model and using the word for what is merely a definition. And, as
> Russell points out, a definition is merely a syntatic substitution,
> so, they don't really exist: it comes back to axioms and rules of
> inference, and nothing else. Which leads me to say, again, that
> concept in math, no, math itself, is bound inside a set of models - it
> doesn't exist in the real world, math is a contrivance of our
> intellect and it only exists inside our own minds, or inside the minds
> of our computers.
>
> Alberto.
That's one view of mathematics. I can think of at least four others.
1) Mathematics is a skill, a means of acquiring knowledge about things and
an aid to manipulating them, and an aid to making decisions. When you use
the correct techniques correctly, you can make good decisions that improve
people's lives. Mistakes can destroy property and kill people. A skilled
craftsman knows the strengths and limitations of his tools and techniqes.
2) Mathematics is a science, a combination of knowledge and concepts
accumulated and built on by the experience and experimentation of
generations of past mathematicians. It is possible to discover new things
about mathematics by experimentation and trial. The most important
difference from other experimental studies is that strict logical
consistency with already accepted results using the rules of logic, instead
of conformity with observation, form the the criteria for acceptance or
rejection of proposed theories.
3) Mathematics is an art, the product of creation of human minds. People may
create new mathematical objects or concepts. Some of these are utilitarian,
some of them are beautiful, some more one than the other, some both
beautiful and utilitarian, some neither. As with any other art, fashion,
taste, and style change with time.
4) Mathematics is a language, a specialized language, or jargon, used to
communicate ideas about numbers and relationships. Its symbolism might as
well be greek, arabic, or chinese to most people. Not only are there
degrees of fluency in mathematics, but mathematics has various dialects, the
language has changed, and there are subfields with even more specialized
jargon.
I could defend any of these views with analogy and quotation from
prominent mathematicians, but I would have a hard time defending any of
them as The One True and Only way to view or teach mathematics and excluding
the others.
Thad Coons
Herman Rubin
So why was this not "practical for the classroom"? High
school students had been learning from this for centuries,
and it was used more than 2000 years ago. Do we need more
proof than this that it is teachable?
In fact, Euclid's students took it with no mathematics
background whatever, as the use of variables had not
yet been invented. They probably could do arithmetic,
but nobody worried about speed or much proficiency.
>>The old Euclid books were used in the standard college
>>preparatory program before. To some extent, they were
>>even teacher proof, as theorems and proofs were carefully
>>explained, but naturally there was nothing teaching
>>students how to go about proving theorems.
>Which was the problem with the SMSG book as well.
Why is this a problem? One CANNOT teach how to prove
theorems. This is not quite true; it could be done for
geometry, but many of the proofs would be so long and
convoluted as to be almost incomprehensible. This is
relatively recent.
>>I use the term
>>"naturally" because this is something we do not know how
>>to do, and probably never will. Creativity can be evoked,
>>but not taught.
>Polya made an effort, and I understood a little of what he said, but it was
>too far above my level at the time to really get it.
I have seen this, but it is not the way I go about it, and
I was aware that I did not do things that way quite early.
>>When the new math was introduced, nobody expected the
>>teachers to know it. What was expected was that teachers
>>could fairly quickly learn what elementary school children
>>can manage. Nobody thought that they would have problems,
>>and if they had any understanding of mathematical concepts,
>>they would not.
>They expected the teachers to learn things without having had any background.
>Other than being able to read somewhat more complex info a little faster, the
>teacher will likely take as long to learn as the kids.
The teachers should be able to do somewhat better, as the
kids did not have the previous acquaintance with algebra
and symbolic notation, and the kids in the early grades did
not even read very well.
For the high school material, there was not that much that
the teachers were not supposed to have already had. The
mathematics in the SMSG books was mainly already required,
in many places even for college admission.
Why anyone would
>expect a teacher, even one just out of college, to learn in a summer course
>any more than a college student would learn in a summer course, baffles me.
Nobody expected that. They were to get the concepts, not
the results of working with them. This should enable them
to be able to use the concepts in reading the textbooks.
Bob LeChevalier
>>>A concept is not a set of words, to be memorized and
>>>regurgitated when requested. It is not a set of facts.
>>>It is not composed of procedures.
>
>>Irrelevant.
>
>NOT irrelevant. Those drilled in facts and procedures
>almost need to start over.
You seem to have a fundamental disagreement with Suppes, since apparently
that was precisely what he was using in his CAI courses used for teaching
bright kids, which you constantly hold up to show that it can be done. Any
course that has kids doing 100-150 problems in a class in one day is
precisely doing drill.
Now you are saying that he in fact could not have been teaching them, and
that they would have almost needed to start over.
So why should we believe you when you say that younger kids can learn the
stuff, when the pre-eminent experimenter used precisely the techniques that
you say cannot work?
>Now what happens if students are not given this type of course?
>It cannot be done for all, as the staff required is too high.
>Even if the book has the derivation, they skip over it, and
>teach just the formulas. We would do better to have the students
>just read the book and ask questions, and not have them teach.
That is because no one has figured out how to teach the derivation. You
think it is because they do not understand the derivation; they know it is
because the kids will have no idea what they are talking about if they
explain it like you did to them. The disconnect, as I said, is that no one
knows how to teach the teachers how to teach this stuff, whether they
understand it or not.
> The
>>teachers learned and understood the first concept and tried to figure out how
>>they could teach it.
>
>You are assuming that the concept was presented as words to
>be memorized. It isn't. You cannot be taught how to teach
>a concept; if the presentation and the exercises do not work,
>you use the understanding of the concept to look at it from
>another angle. And you do not use a time schedule. Having
>a daily "lesson plan" to be followed precisely cannot work to
>teach concepts.
Then you have nothing useful to teach a teacher, because the teachers HAVE TO
have a lesson plan (in some cases there are required to as part of their
employment, in other cases it is merely because they have a certain amount of
time to cover the course and they must get through all of it, which cannot be
done without a lesson plan.
>>Their job is to teach the material directed by the state, whether that
>>teaching does harm or good. If there is a way that the can teach it that
>>does not harm, they want to learn that way. The concepts you are teaching
>>them MUST be presented as a step towards that learning AND the professor
>>needs to make it clear that he understand why they are trying to learn.
>
>This is why we need alternate educational paths. We had students
>who understood 50 years ago; we do not have them now.
Where are they? According to you, they haven't been in the classroom.
Bob LeChevalier
>So learn from the textbook, and ask questions.
I couldn't understand one paragraph of that textbook. One look at
mathematical notation and my eyes glazed over. It made no sense at all to
me.
>Most faculty teaching such a course are willing to answer them, or to
>provide help otherwise.
The concept of asking an adult for help is completely alien to a teenager,
and doubly so to someone like me who had never needed to ask for help from
anyone. And I was still overly impressed by the "Doctor" that we put in
front of their names.
>And, which few students seem to do,
>read the section to be discussed BEFORE coming to class.
That'd be nice, if we had understood the section before.
>>I didn't take college algebra. They discussed induction in high school
>>algebra, but I had no interest in proof, and never figured out why I should
>>care that they used induction to prove this or that.
>
>College algebra used to be standard in high school. It
>usually was part of the second year of algebra.
In other words, what we called Algebra II. I was the kid who they allowed to
do precisely what you call for - move ahead at whatever pace we could. I did;
had no problem doing any of the problems in the textbook including the proofs
except for the ones dealing with limits and infinite series, which I could
not understand at all (and never did).
>So you did not want to know why,
No. I did not CARE about why. For that matter I did not care about anything
other than whatever was explicitly stated in the textbook, because I was
adept at learning and grasping what was explicitly stated. You make the
assumption that most kids are curious to learn. In a sense they are, but
they usually are only curious about things that can be explained in 10
seconds or less: they want answers, data, not the need to think about
answers.
>and you cut yourself off from understanding the basic properties of the integers.
Yes. And if you had put it to me like that, I would have wondered "why the
heck should I care about the basic properties of the integers." The only
purpose of numbers I understood is to get answers to questions of 'how many',
and the only property I cared about was that they were useful tools for
figuring out 'how many'. I still think of mathematics only as a tool, and
"understanding the basic properties of the integers", even when I know what
you mean by those properties, means as little to me as "understanding the
basic properties of a hammer and a saw". Duh, one is heavy and hits things,
and the other is flat with sharp teeth and cuts --- end of story (and yet now
I know that there ARE other properties of hammers and saws, which is why they
come in multiple shapes and sizes - but at age 15 all hammers are the same).
> I understood what they
>>had done, but only as a demonstration; I could prove things like trig
>>identities that were straightforward manipulation with cute substitution
>>tricks, but I don't think I even tried any problems that asked for a proof by
>>induction - how should I know how to do it?
>
>It was presented; that was enough reason to consider not
>just learning it, but understanding it.
The concept of learning anything more than what was explicitly stated in the
book was as of yet unconsidered by me. It was my college English and history
classes that first suggested the possibility that my mind could come up with
useful and meaningful insights above and beyond what was explicitly told to
me in the book. Science and math classes gave the impression that we were
following the trail of mathematicians and scientists who had done it all
before, and the process of becoming like them was to do precisely what the
book told us to do. "Why?" was simply not a question one asked.
>Curiosity may have killed the cat, but not having it is
>deadly for education. Young children have it, but it seems
>to disappear. Could the way the schools approach things
>have anything to do with this?
As I said, young kids have it, and so do older kids. My teenagers are just
as curious now as when they started school. But just as they were then,
their curiosity is shallow. They want a quick answer to their questions that
they do not have to consciously think about. Their minds are working
elsewhere, and the questions they ask are merely to fill a void so that the
world makes enough sense that they can think about whatever it is that is
really on their minds. My daughter constantly teases me because I cannot
give her an answer without a 5 minute (or more) explanation - what she wants
is a 5 WORD explanation, using no words or concepts that she doesn't already
know.
>>Since it wasn't made clear that we should CARE whether they hold or not, it
>>was noise. Students often learn what they are told merely because they are
>>told to. Figuring out why requires them to believe that the reason why is
>>accessible to them and important to them.
>
>Learning is for the FUTURE, not the present. Tying it to what
>is of present interest does great harm.
Kids live in the NOW.
> To most kids, the only such reason
>>is "because the teacher says so, and you'll flunk if you don't".
>
>The schools which teach this should be replaced completely,
>or we cannot get educated people.
Schools don't teach it. That is the way the kids think, and nothing that the
schools can do changes it. Most are far more interested in what they will do
after school than in what they will need to know in the future. My daughter
made more effort in here classes this year ONLY after we made it clear to her
that the alternative was summer school that would force her to miss out on
the sailing that she loves. Even at 16, adulthood is still so far away from
her in her future that she really doesn't think seriously about it, despite
our efforts. (The exception is learning to drive - she DOES take seriously
the comments that I make while driving, and clearly thinks about them - but
learning to drive is something she CARES about).
>>>The one who understands the
>>>concepts sees this type of situation everywhere; algebra
>>>is not just for numbers, and should not be taught that way.
>>>General concepts are easier.
>
>>I first saw that (algebra is not just for numbers) sometime around 1985, 11
>>years after finishing college.
>
>That is why it belongs in first grade, and not just for
>numbers.
I wouldn't have understood it then, and I couldn't explain it to my teenage
kids now in a way that would make sense to them. It takes experience before
you can get the idea that you can add things other than ordinary numbers and
that it will still be meaningful.
lojbab
Bob LeChevalier
>In article <2d8aiu47hdib3tvb2...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>>>The textbooks were not practical for the classroom (I know because I studied
>>>>from the SMSG Geometry text, and also tutored other kids, some quite bright,
>>>>using it).
>>I don't recall that the book I used mentioned Euclid at all, but it most
>>certainly was proofs, proofs and more proofs, seldom with motivation
>>presented for why one would care to prove whatever it was. The lack of
>>connection with algebra or anything else that had come before in math totally
>>baffled kids. Most who made it simply took the attitude that it didn't have
>>to make sense - they were learning it because that is what was in the
>>textbook. Only the unit on vectors near the end of the book had some
>>calculation.
>
>So why was this not "practical for the classroom"? High
>school students had been learning from this for centuries,
>and it was used more than 2000 years ago. Do we need more
>proof than this that it is teachable?
It is studiable, but do the kids learn it. SMSG came about because the
Euclid based books that came before were NOT being understood. The kids were
doing whatever its was that they were told to do, closing the book and
forgetting every last bit of it. I doubt that one kid in a hundred, 5 years
after studying Euclid, could prove a single theorem. With SMSG, they
probably raised it to one kid in ten. The other 90% came away not really
knowing what parallel lines were and unable to calculate the area of a
triangle unless given the formula and the numbers.
>In fact, Euclid's students took it with no mathematics
>background whatever, as the use of variables had not
>yet been invented. They probably could do arithmetic,
>but nobody worried about speed or much proficiency.
And the pinnacle of achievement for most of Euclid's students was probably to
master the material before they reached your age, because there really was
nothing BEYOND Euclid. Even so, we don't read about the students that did
not make it.
>>>The old Euclid books were used in the standard college
>>>preparatory program before. To some extent, they were
>>>even teacher proof, as theorems and proofs were carefully
>>>explained, but naturally there was nothing teaching
>>>students how to go about proving theorems.
>
>>Which was the problem with the SMSG book as well.
>
>Why is this a problem?
The kids come away having learned NOTHING.
>For the high school material, there was not that much that
>the teachers were not supposed to have already had. The
>mathematics in the SMSG books was mainly already required,
>in many places even for college admission.
That which someone had several years ago in a course, and which has never
been used since, is usually GONE, and needs to be totally relearned. The
relearning might be done more quickly, but if you try to teach expecting them
to know it already, you'll lose them the first day. And you'd be equally
foolish to expect that they will have prepared for the class before the first
day by refreshing their memories. I know a few people who would do so, but
not most.
> Why anyone would
>>expect a teacher, even one just out of college, to learn in a summer course
>>any more than a college student would learn in a summer course, baffles me.
>
>Nobody expected that. They were to get the concepts, not
>the results of working with them.
You've said that there are math majors who manage to go through an entire
degree program and don't get the conceptual understanding they need, and
you've also said that you can't teach concepts. So how could you accomplish
it in 9 weeks or less?
lojbab
toto
Rubin) wrote:
>In fact, Euclid's students took it with no mathematics
>background whatever, as the use of variables had not
>yet been invented. They probably could do arithmetic,
>but nobody worried about speed or much proficiency.
Since we actually know little of Euclid's life and who is'
students were, I don't know that you can prove this. One
of the very few anecdotes we have about Euclid is the
following story
http://library.thinkquest.org/22494/stories/Euclid.htm?tqskip1=1&tqtime=0705
:Only two anecdotes about Euclid have come down to us,
:and both are doubtful. In his Eudemiarz Summary, Proclus
:(410-485) tells us that Ptolemy Soter, the first King of Egypt
:and the founder of the Alexandrian Museum, patronized
:the Museum by studying geometry there under Euclid. He
:found the subject difficult and one day asked his teacher if
:there weren't some easier way to learn the material. To this
:Euclid replied, "Oh King, in the real world there are two
:kinds of roads, roads for the common people to travel upon
:and roads reserved for the King to travel upon. In geometry
:there is no royal road."
You seem to make up facts because you want it to be that
way. We don't even know how old the students who were
studying with Euclid were much less how much arithmetic
they knew or if they knew much of the science of the time.
Archimedes was a contemporary of Euclid's students, but
we have no idea if that means they were young adults or
young children.
The other anecdote, btw:
:The second anecdote about Euclid that has come down
:to us is an unreliable but pretty story told by Stobaeus in
:his collection of extracts, sayings, and precepts for his
:son. One of Euclid's students, when he had learned the
:first proposition, asked his teacher, "But what is the good
:of this and what shall I get by learning these things?"
:Thereupon Euclid called a slave and said, "Give this fellow
:a penny, since he must make gain from what he learns. "
Dorothy
toto
Rubin) wrote:
>So learn from the textbook, and ask questions. Most faculty
>teaching such a course are willing to answer them, or to
>provide help otherwise. And, which few students seem to do,
>read the section to be discussed BEFORE coming to class.
>Professors are not equipped with magic wands which enable the
>ideas to penetrate the major barrier to the transmission of
>knowledge, 1/4 inch of human skull.
There is a big problem with this because reading a math text is
NOT the same as reading prose and no one teaches kids how to
read them.
toto
Rubin) wrote:
>>I never, in math department courses, or any other math courses,
>>figured out why I needed to know how the mathematicians had
>>proven it, rather than merely THAT they had proven it, and I had
>>no curiosity for the question.
>
>Curiosity may have killed the cat, but not having it is
>deadly for education. Young children have it, but it seems
>to disappear. Could the way the schools approach things
>have anything to do with this?
Yes, I agree that it does. Young children love learning for the
sake of learning. When you add competition and grades to the
mix, when you try to separate subjects from what makes sense
in the real world, when you don't connect the dots of learning,
when you insist on trying to push them into things that are
beyond them *or* to keep them back from things that they can
do, then kids begin to believe it's useless and they turn off their
brains.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>>A concept is not a set of words, to be memorized and
>>>>regurgitated when requested. It is not a set of facts.
>>>>It is not composed of procedures.
>>>Irrelevant.
>>NOT irrelevant. Those drilled in facts and procedures
>>almost need to start over.
>You seem to have a fundamental disagreement with Suppes, since apparently
>that was precisely what he was using in his CAI courses used for teaching
>bright kids, which you constantly hold up to show that it can be done. Any
>course that has kids doing 100-150 problems in a class in one day is
>precisely doing drill.
This is one of the problems with computer classes, not
just CAI. It is very hard to get the feedback. It takes
a human to make judgments.
However, is it drill in facts and procedures? One can
give a large number of problems which are not that.
>Now you are saying that he in fact could not have been teaching them, and
>that they would have almost needed to start over.
If the concepts are presented first, and are required
to be used, excessive drill is only wasteful and boring.
Suppes is one of the authors of a paper which shows that
mathematical concept formation requires no time spent on
exercises after it is learned, and no benefit is gained.
>So why should we believe you when you say that younger kids can learn the
>stuff, when the pre-eminent experimenter used precisely the techniques that
>you say cannot work?
>>Now what happens if students are not given this type of course?
>>It cannot be done for all, as the staff required is too high.
>>Even if the book has the derivation, they skip over it, and
>>teach just the formulas. We would do better to have the students
>>just read the book and ask questions, and not have them teach.
>That is because no one has figured out how to teach the derivation.
This is false. But one cannot expect the type of "learning"
which is done by drill, although if the students were not
taught by memorization and algorithm, one probably would get
better results. The results were better 50 years ago, and
this includes those using the GI Bill.
You
>think it is because they do not understand the derivation; they know it is
>because the kids will have no idea what they are talking about if they
>explain it like you did to them. The disconnect, as I said, is that no one
>knows how to teach the teachers how to teach this stuff, whether they
>understand it or not.
They disconnect because they have never had a course which
used derivations, and expected them to use them. This
belongs well before college, and even before high school.
Teaching why should be the rule, not the exception.
>> The
>>>teachers learned and understood the first concept and tried to figure out how
>>>they could teach it.
>>You are assuming that the concept was presented as words to
>>be memorized. It isn't. You cannot be taught how to teach
>>a concept; if the presentation and the exercises do not work,
>>you use the understanding of the concept to look at it from
>>another angle. And you do not use a time schedule. Having
>>a daily "lesson plan" to be followed precisely cannot work to
>>teach concepts.
>Then you have nothing useful to teach a teacher, because the teachers HAVE TO
>have a lesson plan (in some cases there are required to as part of their
>employment, in other cases it is merely because they have a certain amount of
>time to cover the course and they must get through all of it, which cannot be
>done without a lesson plan.
>>>Their job is to teach the material directed by the state, whether that
>>>teaching does harm or good. If there is a way that the can teach it that
>>>does not harm, they want to learn that way. The concepts you are teaching
>>>them MUST be presented as a step towards that learning AND the professor
>>>needs to make it clear that he understand why they are trying to learn.
>>This is why we need alternate educational paths. We had students
>>who understood 50 years ago; we do not have them now.
>Where are they? According to you, they haven't been in the classroom.
The educationists' idea of teaching all children of the
same age the same material, and reducing the classes to the
least common denominator, did not reach the high schools
much until after WWII. The students of 50 years ago had
Euclid, and most had a "college algebra" course in high
school which spent time on the use of induction.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>So learn from the textbook, and ask questions.
...............
>>And, which few students seem to do,
>>read the section to be discussed BEFORE coming to class.
>That'd be nice, if we had understood the section before.
School should be a place to learn. If you do not understand,
ask questions! This should be encouraged from the beginning,
not put off. The only stupid question is the one which is
not asked.
>>>I didn't take college algebra. They discussed induction in high school
>>>algebra, but I had no interest in proof, and never figured out why I should
>>>care that they used induction to prove this or that.
>>College algebra used to be standard in high school. It
>>usually was part of the second year of algebra.
>In other words, what we called Algebra II.
In the old day, courses were by semesters, not years.
I was the kid who they allowed to
>do precisely what you call for - move ahead at whatever pace we could. I did;
>had no problem doing any of the problems in the textbook including the proofs
>except for the ones dealing with limits and infinite series, which I could
>not understand at all (and never did).
Limits are NEEDED to understand the "infinite decimals"
usually taught around 6th grade. Without limits, there
is no way to understand infinite series. You were
supposedly there to learn; when you did not understand,
and you knew it, you did nothing about it.
>>So you did not want to know why,
>No. I did not CARE about why. For that matter I did not care about anything
>other than whatever was explicitly stated in the textbook, because I was
>adept at learning and grasping what was explicitly stated.
You stated that you did not grasp what was explicitly stated,
or you could have done those problems.
You make the
>assumption that most kids are curious to learn. In a sense they are, but
>they usually are only curious about things that can be explained in 10
>seconds or less: they want answers, data, not the need to think about
>answers.
And nothing important can be taught that way. It is not
the answers which are important; it is asking the questions,
and then figuring out how to get from there to the answer.
So you were a casualty of concentrating on memorization and
routine, rather than understanding, which very often makes
it unnecessary to memorize.
>>and you cut yourself off from understanding the basic properties of the integers.
>Yes. And if you had put it to me like that, I would have wondered "why the
>heck should I care about the basic properties of the integers." The only
>purpose of numbers I understood is to get answers to questions of 'how many',
>and the only property I cared about was that they were useful tools for
>figuring out 'how many'. I still think of mathematics only as a tool, and
>"understanding the basic properties of the integers", even when I know what
>you mean by those properties, means as little to me as "understanding the
>basic properties of a hammer and a saw".
So you deprived yourself of the understanding needed to
make good use of the tools.
Duh, one is heavy and hits things,
>and the other is flat with sharp teeth and cuts --- end of story (and yet now
>I know that there ARE other properties of hammers and saws, which is why they
>come in multiple shapes and sizes - but at age 15 all hammers are the same).
That is why you need to get the understanding BEFORE age
15. Once you get in the rut, it is hard to get out.
>> I understood what they
>>>had done, but only as a demonstration; I could prove things like trig
>>>identities that were straightforward manipulation with cute substitution
>>>tricks, but I don't think I even tried any problems that asked for a proof by
>>>induction - how should I know how to do it?
>>It was presented; that was enough reason to consider not
>>just learning it, but understanding it.
>The concept of learning anything more than what was explicitly stated in the
>book was as of yet unconsidered by me.
But it was explicitly stated in the book.
It was my college English and history
>classes that first suggested the possibility that my mind could come up with
>useful and meaningful insights above and beyond what was explicitly told to
>me in the book. Science and math classes gave the impression that we were
>following the trail of mathematicians and scientists who had done it all
>before, and the process of becoming like them was to do precisely what the
>book told us to do. "Why?" was simply not a question one asked.
The why was there; you just ignored it. You wanted
simple answers, and fooled yourself into thinking that
this was all that mattered.
>>Curiosity may have killed the cat, but not having it is
>>deadly for education. Young children have it, but it seems
>>to disappear. Could the way the schools approach things
>>have anything to do with this?
>As I said, young kids have it, and so do older kids. My teenagers are just
>as curious now as when they started school. But just as they were then,
>their curiosity is shallow. They want a quick answer to their questions that
>they do not have to consciously think about. Their minds are working
>elsewhere, and the questions they ask are merely to fill a void so that the
>world makes enough sense that they can think about whatever it is that is
>really on their minds. My daughter constantly teases me because I cannot
>give her an answer without a 5 minute (or more) explanation - what she wants
>is a 5 WORD explanation, using no words or concepts that she doesn't already
>know.
So this has to start early, not late, or the person's mind
will be turned off from the real explanation.
>>>Since it wasn't made clear that we should CARE whether they hold or not, it
>>>was noise. Students often learn what they are told merely because they are
>>>told to. Figuring out why requires them to believe that the reason why is
>>>accessible to them and important to them.
>>Learning is for the FUTURE, not the present. Tying it to what
>>is of present interest does great harm.
>Kids live in the NOW.
What is the purpose of education? The teachers who teach
only for the NOW are at least weakening, if not destroying,
their minds. What are YOU, as a parent, doing about it?
>> To most kids, the only such reason
>>>is "because the teacher says so, and you'll flunk if you don't".
>>The schools which teach this should be replaced completely,
>>or we cannot get educated people.
>Schools don't teach it.
They do. Those who have only learned themselves by
memorization and regurgitation can do nothing else.
That is the way the kids think, and nothing that the
>schools can do changes it. Most are far more interested in what they will do
>after school than in what they will need to know in the future. My daughter
>made more effort in here classes this year ONLY after we made it clear to her
>that the alternative was summer school that would force her to miss out on
>the sailing that she loves. Even at 16, adulthood is still so far away from
>her in her future that she really doesn't think seriously about it, despite
>our efforts. (The exception is learning to drive - she DOES take seriously
>the comments that I make while driving, and clearly thinks about them - but
>learning to drive is something she CARES about).
>>>>The one who understands the
>>>>concepts sees this type of situation everywhere; algebra
>>>>is not just for numbers, and should not be taught that way.
>>>>General concepts are easier.
>>>I first saw that (algebra is not just for numbers) sometime around 1985, 11
>>>years after finishing college.
>>That is why it belongs in first grade, and not just for
>>numbers.
>I wouldn't have understood it then, and I couldn't explain it to my teenage
>kids now in a way that would make sense to them. It takes experience before
>you can get the idea that you can add things other than ordinary numbers and
>that it will still be meaningful.
Variables are a simple linguistic tool. They make it
possible to communicate clearly about almost anything.
toto
Rubin) wrote:
>School should be a place to learn. If you do not understand,
>ask questions! This should be encouraged from the beginning,
>not put off. The only stupid question is the one which is
>not asked.
Boy did you hit the nail on the head here. The biggest problem
that older student's have is that they are afraid to ask questions
because they think it will make the other students laugh at them
for being stupid.
But that's part of the reason why teachers have to also teach
the social values of acceptance of questioning in school in the
first place. If we don't, then the student's will continue *not* to
ask out of fear of embarrassment. That is where the student's
self-esteem and the classroom atmosphere of acceptance
come into play.
toto
Rubin) wrote:
>Limits are NEEDED to understand the "infinite decimals"
>usually taught around 6th grade. Without limits, there
>is no way to understand infinite series.
I agree with this and I think there are other places where
and understanding of limits is helpful if not essential.
>You were supposedly there to learn; when you did not
>understand, and you knew it, you did nothing about it.
Most children would not exactly know what to do about it
especially if the teachers were letting him go on his own
and not really checking on him
Bob LeChevalier
>In article <uq3ciuc2guui2u5qc...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>>>A concept is not a set of words, to be memorized and
>>>>>regurgitated when requested. It is not a set of facts.
>>>>>It is not composed of procedures.
>
>>>>Irrelevant.
>
>>>NOT irrelevant. Those drilled in facts and procedures
>>>almost need to start over.
>
>>You seem to have a fundamental disagreement with Suppes, since apparently
>>that was precisely what he was using in his CAI courses used for teaching
>>bright kids, which you constantly hold up to show that it can be done. Any
>>course that has kids doing 100-150 problems in a class in one day is
>>precisely doing drill.
>
>This is one of the problems with computer classes, not
>just CAI. It is very hard to get the feedback. It takes
>a human to make judgments.
>
>However, is it drill in facts and procedures? One can
>give a large number of problems which are not that.
Given that is has been YOU who's been touting Suppes' success in teaching
younger kids math concepts that the teachers supposedly couldn't learn (I've
found no evidence that he or anyone else tried to teach teachers using the
same CAI methods, BTW), I would have presumed that you had some idea what it
was that had been taught and how they taught it.
>>Now you are saying that he in fact could not have been teaching them, and
>>that they would have almost needed to start over.
>
>If the concepts are presented first, and are required
>to be used, excessive drill is only wasteful and boring.
>Suppes is one of the authors of a paper which shows that
>mathematical concept formation requires no time spent on
>exercises after it is learned, and no benefit is gained.
Of course the question is how you tell when a concept has been learned, since
you've defined it as not being learned until the student can apply the
concept years later.
>>>Now what happens if students are not given this type of course?
>>>It cannot be done for all, as the staff required is too high.
>>>Even if the book has the derivation, they skip over it, and
>>>teach just the formulas. We would do better to have the students
>>>just read the book and ask questions, and not have them teach.
>
>>That is because no one has figured out how to teach the derivation.
>
>This is false.
Read it again slowly. You have taught the teachers the derivation, but not a
method of teaching the derivation to students that are well behind the
teacher's level of intellectual maturity (since few teachers are teaching
ONLY kids who are as smart as the teachers, this would seem obvious). It is
not necessarily the case that someone understanding something can figure out
how to effectively teach it.
>But one cannot expect the type of "learning"
>which is done by drill, although if the students were not
>taught by memorization and algorithm, one probably would get
>better results. The results were better 50 years ago, and
>this includes those using the GI Bill.
The people who started the New Math era didn't seem to think so.
> You
>>think it is because they do not understand the derivation; they know it is
>>because the kids will have no idea what they are talking about if they
>>explain it like you did to them. The disconnect, as I said, is that no one
>>knows how to teach the teachers how to teach this stuff, whether they
>>understand it or not.
>
>They disconnect because they have never had a course which
>used derivations, and expected them to use them.
What would a course use derivations for, except to do more derivations?
I had courses that used derivations, and I never saw the point. It was
mental masturbation to prove stuff that had already proven, when I had no
reason to believe I would ever be doing something "real" that would involve
proving something new. And if I, the top math student in the school, did not
see it, how would the other students see it?
>This
>belongs well before college, and even before high school.
>Teaching why should be the rule, not the exception.
They tried to teach us why. A derivation does not teach us why.
>>>>Their job is to teach the material directed by the state, whether that
>>>>teaching does harm or good. If there is a way that the can teach it that
>>>>does not harm, they want to learn that way. The concepts you are teaching
>>>>them MUST be presented as a step towards that learning AND the professor
>>>>needs to make it clear that he understand why they are trying to learn.
>
>>>This is why we need alternate educational paths. We had students
>>>who understood 50 years ago; we do not have them now.
>
>>Where are they? According to you, they haven't been in the classroom.
>
>The educationists' idea of teaching all children of the
>same age the same material, and reducing the classes to the
>least common denominator, did not reach the high schools
>much until after WWII.
Grade levels based on age started long before WW II.
>The students of 50 years ago had Euclid,
and apparently did not understand it, because it had been watered down by
teachers who likewise did not understand it, which was (per the quotes I
cited), why the New Math effort was initiated.
>and most had a "college algebra" course in high
>school which spent time on the use of induction.
No, most of them did not. Back in 1909-10, when only 6% of people completed
high school, only 56.9% of kids took a year of Algebra, 30.9% took Geometry,
and 1.9% took Trig (which per your comments probably also included College
Algebra). One can hardly call only 2% of 6% "most".
The percentage taking Algebra dropped as the number attending high school
grew, falling to around 25% in 1955, when only 11% took Geometry, but the
number taking Trig/College Algebra was up to 2.6%, which still could not be
called "most", nor even "many".
lojbab
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <2d8aiu47hdib3tvb2...@4ax.com>,
>>Bob LeChevalier <loj...@lojban.org> wrote:
.................
>It is studiable, but do the kids learn it. SMSG came about because the
>Euclid based books that came before were NOT being understood. The kids were
>doing whatever its was that they were told to do, closing the book and
>forgetting every last bit of it. I doubt that one kid in a hundred, 5 years
>after studying Euclid, could prove a single theorem. With SMSG, they
>probably raised it to one kid in ten. The other 90% came away not really
>knowing what parallel lines were and unable to calculate the area of a
>triangle unless given the formula and the numbers.
The SMSG geometry book was very much like the Euclid books,
but with the gaps in assumptions observed removed, and a
small number of additional modifications made. It was not
written because Euclid was that poorly understood, and was
probably the least modified books from the previous.
>>In fact, Euclid's students took it with no mathematics
>>background whatever, as the use of variables had not
>>yet been invented. They probably could do arithmetic,
>>but nobody worried about speed or much proficiency.
>And the pinnacle of achievement for most of Euclid's students was probably to
>master the material before they reached your age, because there really was
>nothing BEYOND Euclid. Even so, we don't read about the students that did
>not make it.
If you think that, you are mistaken. They did know a fair
amount of trigonometry, and also conic sections, of course
not in rectangular coordinates. Euclid's _Elements_ also
included elementary number theory, such as the Euclidean
algorithm for the greatest common divisor, the proof of an
infinite number of primes, uniqueness of prime factorization,
the irrationality of the square root of 2, etc.
There was a knowledge of trigonometry and spherical geometry,
and the evaluation of areas of various figures. Archimedes,
around that time, came up with a justification for the
circumference of a circle and the area of a sphere, and
derived the usual formulas involving pi, as well as bounds
for that requiring half-angle trigonometric formulas.
There were other curves the Greeks knew, such as the directrix,
which could be used to divide angles arbitrarily and get pi
in terms of the limit of the curve at the horizontal axis.
>>>>The old Euclid books were used in the standard college
>>>>preparatory program before. To some extent, they were
>>>>even teacher proof, as theorems and proofs were carefully
>>>>explained, but naturally there was nothing teaching
>>>>students how to go about proving theorems.
>>>Which was the problem with the SMSG book as well.
>>Why is this a problem?
>The kids come away having learned NOTHING.
No, they came away with an understanding of proof, and
geometry based on it.
>>For the high school material, there was not that much that
>>the teachers were not supposed to have already had. The
>>mathematics in the SMSG books was mainly already required,
>>in many places even for college admission.
>That which someone had several years ago in a course, and which has never
>been used since, is usually GONE, and needs to be totally relearned.
Some of the details might, but if the ideas need to be
relearned, the student did not learn the course. This is
even worse now, with only details and methods presented.
The
>relearning might be done more quickly, but if you try to teach expecting them
>to know it already, you'll lose them the first day. And you'd be equally
>foolish to expect that they will have prepared for the class before the first
>day by refreshing their memories. I know a few people who would do so, but
>not most.
>> Why anyone would
>>>expect a teacher, even one just out of college, to learn in a summer course
>>>any more than a college student would learn in a summer course, baffles me.
>>Nobody expected that. They were to get the concepts, not
>>the results of working with them.
>You've said that there are math majors who manage to go through an entire
>degree program and don't get the conceptual understanding they need, and
>you've also said that you can't teach concepts. So how could you accomplish
>it in 9 weeks or less?
What I said is that they almost do not see the concepts.
Did you understand limits after the calculus course? If
not, the course only prepared you to be a calculator.
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On 5 Jul 2002 15:01:45 -0500, hru...@odds.stat.purdue.edu (Herman
>Rubin) wrote:
>>In fact, Euclid's students took it with no mathematics
>>background whatever, as the use of variables had not
>>yet been invented. They probably could do arithmetic,
>>but nobody worried about speed or much proficiency.
>Since we actually know little of Euclid's life and who is'
>students were, I don't know that you can prove this. One
>of the very few anecdotes we have about Euclid is the
>following story
>http://library.thinkquest.org/22494/stories/Euclid.htm?tqskip1=1&tqtime=0705
>:Only two anecdotes about Euclid have come down to us,
>:and both are doubtful. In his Eudemiarz Summary, Proclus
>:(410-485) tells us that Ptolemy Soter, the first King of Egypt
>:and the founder of the Alexandrian Museum, patronized
>:the Museum by studying geometry there under Euclid. He
>:found the subject difficult and one day asked his teacher if
>:there weren't some easier way to learn the material. To this
>:Euclid replied, "Oh King, in the real world there are two
>:kinds of roads, roads for the common people to travel upon
>:and roads reserved for the King to travel upon. In geometry
>:there is no royal road."
This has also been said about Aristotle teaching Alexander,
much earlier.
>You seem to make up facts because you want it to be that
>way. We don't even know how old the students who were
>studying with Euclid were much less how much arithmetic
>they knew or if they knew much of the science of the time.
>Archimedes was a contemporary of Euclid's students, but
>we have no idea if that means they were young adults or
>young children.
>The other anecdote, btw:
>:The second anecdote about Euclid that has come down
>:to us is an unreliable but pretty story told by Stobaeus in
>:his collection of extracts, sayings, and precepts for his
>:son. One of Euclid's students, when he had learned the
>:first proposition, asked his teacher, "But what is the good
>:of this and what shall I get by learning these things?"
>:Thereupon Euclid called a slave and said, "Give this fellow
>:a penny, since he must make gain from what he learns. "
This is quite similar to what is being said about today's
students. BTW, Euclid's first proposition is quite simple;
it is to construct an equilateral triangle with a given side.
The reason I know this is that a high school teacher did not
think much of the second proposition, which I see to be an
excellent pedagogical argument, as it puts lots of things
together to get one result, with each of the steps being
quite straightforward.
The old textbooks left out much of this, to get the coverage
in one year.
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On 5 Jul 2002 14:45:50 -0500, hru...@odds.stat.purdue.edu (Herman
>Rubin) wrote:
>>So learn from the textbook, and ask questions. Most faculty
>>teaching such a course are willing to answer them, or to
>>provide help otherwise. And, which few students seem to do,
>>read the section to be discussed BEFORE coming to class.
>>Professors are not equipped with magic wands which enable the
>>ideas to penetrate the major barrier to the transmission of
>>knowledge, 1/4 inch of human skull.
>There is a big problem with this because reading a math text is
>NOT the same as reading prose and no one teaches kids how to
>read them.
This is why mathematical and other precise subjects need to
be taught early. Reading fiction does not teach someone
how to learn.
But they can read the texts for spelling, arithmetic, and
geography even in the early grades. This makes reading a
tool for learning subject matter, not just literature.
toto
<loj...@lojban.org> wrote:
>>This
>>belongs well before college, and even before high school.
>>Teaching why should be the rule, not the exception.
>
>They tried to teach us why. A derivation does not teach us why.
I have to disagree with you here, Bob. In fact, the derivation
goes to the *why* on a deep level. If you understand the why,
then you can derive the formula when you need it. If you do
not understand, seeing the derivation gives you the chance
to get at the underlying why the formula works.
toto
<loj...@lojban.org> wrote:
A little support for your position here Bob
http://www.edweek.org/sreports/tc99/articles/screening.htm
:*Gerry Solomon, an educational consultant who coordinates software
:*reviews for the North Carolina Department of Public Instruction,
:*agrees. "The best software takes more problem-solving, more thinking
:*skills for the activities," she says. "The students become engaged with
:*doing something with the information presented."
:*
:*But software that includes plenty of practice for the student can also
:*be high-quality, argues Patrick Suppes, who co-founded Computer
:*Curriculum Corp. in 1967 and served as the company's chief executive
:*officer from 1967 to 1990. The Sunnyvale, Calif.-based CCC specializes
:*in a comprehensive package of software--generally known as an
:*integrated-learning system--that not only teaches foundational skills
:*but assesses and tracks student progress.
:*
:*"People who don't recognize the need for practice are only romantics
:*out of touch with the real world," says Suppes, now a professor of
:*philosophy emeritus at Stanford University. "Can you imagine learning
:*how to play basketball by only listening to a lecture and participating
:*in discussions with critical thinking? I would consider it an example of
:*poor quality to offer a student software in mathematics and no
:*opportunity for practice."
Dorothy
:*
<snip>
Thad Coons
> >That's one view of mathematics. I can think of at least four others.
>
> There's more than those. But the problem is, here we are in the
> twentieth first century, deep inside the computer revolution. There's
> a big dichotomy, before computers and after computers. The points of
> view you mention may have been ok before 1990 or so, but things are
> happening pretty fast today.
At least four others is what I said. The first view emphasizes the
tool-like, utilitarian view of mathematics. Computers are also different
things to different people, but in this particular context I consider
paper-and-pencil methods of mathematics as its hand tools, while computers
are its power tools and heavy machinery.
Just as power tools don't replace the attitude of craftsmanship or
substitute for the responsible judgment of the engineer, so computers
neither replace applied mathematicians nor relieve them of responsibility to
use the techniques that will most usefully describe reality.
> >2) Mathematics is a science, a combination of knowledge and concepts
> >accumulated and built on by the experience and experimentation of
> >generations of past mathematicians.
>
> Which is, basically, a tautology. I could say this about math no
> matter which other philosophy of mathematics I subscribe to.
In certain respects, the cumulative progress in mathematics is more like
progress in physics and chemistry than it is like literary criticism or
philosophy.
> >3) Mathematics is an art, the product of creation of human minds.
>
> Which is basically how I see it. Creation: inside our own minds.
Also art because there are those who love the abstract beauty of
mathematical ideas for aesthetic reasons unrelated to their utility.
> >4) Mathematics is a language, a specialized language, or jargon, used to
> >communicate ideas about numbers and relationships.
>
> Now, languages are the realm of computer science
Not exclusively, not while people still speak Chinese, Swahili, or even that
eminent example of consistent rational design, English. Computer languages
are still highly restrictive compared to natural language and they are even
a subset of mathematical language. Things I can say quite easily in English
require elaborate circumlocution in, say C++.
> and we're fast
> learning that there are multiple ways of defining languages. So, to
> say that math is "a language" is, as I see it, very restrictive.
Arithmetic begins with number names, which are words, and symbols, which
are writing. Geometry begins with words for shapes and figures that can be
drawn, and certain relationships among those shapes and figures. If you look
at the history of any mathematical concept, you can trace it ultimately back
to roots in language, even if it is a couple of generations removed.
Mathematics has been subject to the same processes that other languages have
been, including linguistic change. It's not a single, monolithic language by
any means; there are numerous dialects of it.
I'd love to see a historical linguist apply his scholarly tools to
mathematical notation and explain why four different calculus textbooks use
three different notations for the limit and about five for the derivative;
or why symbolic logic has several different systems of notation, or what the
proliferation of systems of notation in higher mathematics, systems
incomprehensible to nonspecialists, says about communication in the
mathematical community and academic specialization in general.
> Still, one thing I don't think one can say is, that mathematics really
> "exists" outside itself, that is, in objective nature. So, whatever
> tack we take to learn it, we're bound inside its own models. And in
> the end, what we non-mathematicians want out of math is that clay
> thing, that ability to model, and that ability to reason in the model
> instead of reasoning in problem space: not unlike the engineer who
> builds a lab model of the Bay of Fundy, because it's beyond his grasp
> to experiment with nature in the raw.
> The logicist view of math - and unlike those you listed, I'd say that
> the other ways of looking at them are, for example, Brouwer's
> intuitionism, or Hilbert's formalism, or even Church's lambda
> calculus. All of them share to different extents one common trait,
> though, the locking within the bounds of the model itself, and the
> dissociation of mathematical thought from objective reality.
Logic and these other ways of looking at mathematics use highly abstract
mathematical concepts and notation and are inaccessible to the layman. They
also suffer from the same problems any language has when it becomes its own
subject matter. Self reference and the distinction between use and mention
raise inherently unavoidable problems that don't appear when the population
of Egypt or the factors of 79355 are the subject matter.
> And from a user point of view, say, an engineer, or a computer man -
> and I'm both - what comes out very intensely is that mathematics
> allows us to leave problem space, come to conclusions by modeling
> technique alone, take the results, then feed them back into our
> perception of reality, refine our model, compute again, and so on. But
> the dissociation of model and reality is always there.
>
> And concept is fluid: I define addition the way I want, and addition
> in the good old mathematics ain't quite the addition we use, say,
> inside our modern computers. Today we have, for example, object
> orientation: we define our own operations, our own objects, our own
> language, our own syntax and semantics. One student of mine, long ago,
> put it in its proper plane: PROGRAMMING IS ABOUT CREATING A LANGUAGE
> THAT BEST EXPRESS THE PROBLEM SPACE WE'RE DEALING WITH - and that
> immediately puts that concept of "math is a language" in check. Math
> ain't just a language, because I can superimpose to it whatever
> language I will, and the basics don't change !
The same claim you have made for mathematics, that it is a symbolic
representation of reality, is similar to what what general semantics has
claimed about language itself: The map is not the territory; the language is
not the thing it describes.
I've gained useful insights from each of these views, and I don't think any
one of them is sufficient by itself. I suppose what irks me is the idea that
mathematics is a self-contained thing unto itself, independent of and
superior to all other forms of knowledge. Mathematicians are necessary in a
modern civilized society, but they are not the High Priests of the Exalted
Temple of Wisdom, Guardians of the Mysteries,
Servants of Nihil From Which All Things Come.
Thad Coons
Bob LeChevalier
>Bob LeChevalier <loj...@lojban.org> said:
>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>So learn from the textbook, and ask questions.
>>
>>I couldn't understand one paragraph of that textbook. One look at
>>mathematical notation and my eyes glazed over. It made no sense at all to
>>me.
>
>the book ? Or maybe you didn't have the adequate level of skill to be
>able to read mathematical notation in a more efficient way ? Or maybe
>you lacked the willpower to put in the time to decode the mathematical
>notation ?
All of the above. No one in all my years of schooling EVER taught me to read
mathematical notation efficiently, or even to read it at all. We were taught
what the symbols meant, and (sudden insight!) the situation with math
notation for me and most students is precisely the same as a kid who learns
via phonics to decode the symbols on the page and assemble them into words,
but has no idea what many of the words mean, much less what entire phrases
and sentences mean. We never learn to read math better than the typical 1st
grader reads English prose. It is presumed that once we know the symbols, we
can figure out the rest.
>I believe I already passed this on, but I will repeat. It's from the
>preface of Sheldon Axler's "Linear Algebra Done Right" book:
>
>=====================
>"You cannot expect to read mathematics the way you read a novel. If
>you zip through a page in less than an hour, you are probably going
>too fast. When you encounter the phrase "as you should verify", you
>should indeed do the verification, which will usually require some
>writing on your part. When steps are left out, you need to supply the
>missing pieces. You should ponder and internalize each definition. For
>each theorem, you should seek examples to show why each hypothesis is
>necessary."
>=====================
All well and good that this is what "should be", but K12 students are NOT
going to spend an hour reading a page, and will not verify anything (or do
anything else) that they are not specifically assigned to verify by the
teacher. A K12 teacher who expects the kids to spend more than an hour a day
on only ONE subject will probably not keep his job. High school teachers
today would probably be happy if kids consistently spent an hour a day total
outside of class on their homework for ALL their classes, with the ideal high
school kid perhaps spending 2 hours per day on homework.
>>No. I did not CARE about why. For that matter I did not care about anything
>>other than whatever was explicitly stated in the textbook, because I was
>>adept at learning and grasping what was explicitly stated. You make the
>>assumption that most kids are curious to learn. In a sense they are, but
>>they usually are only curious about things that can be explained in 10
>>seconds or less: they want answers, data, not the need to think about
>>answers.
>
>If you don't care for proof, learning becomes parrotlike: you take
>statements at face value, and you just repeat them. Proofs set up the
>background for theorems, they give a justification for the result, and
>they often develop additional material on the side that is important
>to be digested if the student wants to understand what the theorem is
>really all about. It is often the case that the proof conveys more
>information than the theorem itself. So, if all you want is to read
>something and grasp it within ten seconds, mathematics is indeed the
>wrong field to study.
I may understand this now, but at age 16 when I had had a history of grasping
the math I had been taught in 10 seconds rather than an hour a page, it was
not something I was prepared to accept even if someone had bothered to tell
me.
>>>and you cut yourself off from understanding the basic properties of the integers.
>>
>>Yes. And if you had put it to me like that, I would have wondered "why the
>>heck should I care about the basic properties of the integers." The only
>>purpose of numbers I understood is to get answers to questions of 'how many',
>>and the only property I cared about was that they were useful tools for
>>figuring out 'how many'.
>
>But that requires knowing the basic properties of the integers. One
>reason we study math is to replace naive reasoning with mathematical
>reasoning.
But of course no one ever said that. The purpose of studying math was to
solve problems that used math. That there was a distinction between naive
reasoning and mathematical reasoning never was stated. We were taught that
there was this thing called "formal proof" where you write out all the steps,
but that was something one did only when proving theorems, which nobody cared
about anyway.
Watching my own kids work, I recognize now that the less skilled math
students probably had an advantage over me, in fact. Most of them were used
to having to write out all the steps in order to solve a simple addition
problem or later, a linear equation in one unknown. Since I could do all
that in my head, I tended to consider that all the math worth doing was the
stuff I could do in my head, or at most writing one intermediate step as a
memory aid. My son, who has a writing problem, is similar to me, except that
he often doesn't get the right answer when he does it in his head, so he not
only doesn't see the need for rigor, he CANNOT do a rigorous problem. My
daughter used to write the steps out in full for every problem, but long
before 7th grade this was taking her hours per day, and she got to the point
where she wasn't willing to spend that much time on each day's homework.
>It often happens that we can reach the same point using
>naive reasoning, BUT THE WHOLE POINT OF LEARNING MATH IS TO LEARN HOW
>TO GET THERE USING MATHEMATICAL THOUGHT, AND NOT NAIVE THOUGHT - even
>when naive thought seems to be more straightforward. So, maybe you can
>answer some of thoes "how many" questions without engaging in math
>that's too serious for your taste, but that's flaunting the purpose of
>studying the stuff!
But no one ever tells kids that math is any more than finding the answers to
"how many" questions by whatever means works (with the methods that are
taught being most likely to get the correct answer in the quickest amount of
time). The idea that math reasoning is different or any more rigorous than
any other type of reasoning is never presented to the kids, and I've found
now that I know this is the case, that >I< have been unable to present it to
MY kids.
We have not learned how to teach kids that naive reasoning is not "good
enough". And by the time they have learned it, the math is usually way over
their head because they had run through years of classes being content with
naive reasoning so long as it got the right answer, which it usually does at
the lower levels.
>> I still think of mathematics only as a tool, and
>>"understanding the basic properties of the integers", even when I know what
>>you mean by those properties, means as little to me as "understanding the
>>basic properties of a hammer and a saw". Duh, one is heavy and hits things,
>>and the other is flat with sharp teeth and cuts --- end of story (and yet now
>>I know that there ARE other properties of hammers and saws, which is why they
>>come in multiple shapes and sizes - but at age 15 all hammers are the same).
>
>But mathematics as a tool is more like a computer than like a hammer
>or a saw.
Most 15 year olds also see a computer as a tool, and all computers look alike
to them.
>It is, indeed, more like a general purpose tool-building
>toolkit, one that you can reconfigure in a myriad of different tools
>that you can use for different purposes at different times.
That sounds like the Unix way of looking at computers. Most people use one
or two basic functions of their computer - a browser, an email program, maybe
an editor or a spreadsheet, and sometimes even these are integrated so as to
all work like one slightly more complex tool. The concept of reconfiguring a
program as a tool to do a different function is something most kids never
learn.
You object to bringing the computer into the classroom, but when the teachers
and the students are not even proficient at these simple functions, it never
occurs to them to do anything more complex.
>>The concept of learning anything more than what was explicitly stated in the
>>book was as of yet unconsidered by me. It was my college English and history
>>classes that first suggested the possibility that my mind could come up with
>>useful and meaningful insights above and beyond what was explicitly told to
>>me in the book. Science and math classes gave the impression that we were
>>following the trail of mathematicians and scientists who had done it all
>>before, and the process of becoming like them was to do precisely what the
>>book told us to do. "Why?" was simply not a question one asked.
>
>Yet, when many students have problems with math, that "why" question
>is a nagging constant. How many times I heard that "why" question from
>people who just lack the prerequisite to learn the stuff in a proper
>way?
And the reason they lack the prerequisite is that nobody has added that
prerequisite to the curriculum. You will likely not find ANYWHERE in a K/12
math textbook that mathematics is fundamentally different from other academic
subjects in terms of rigor, and you'll never find anything like what you just
said.
>You see, your attitude seems to be, "I love naive logic and I don't
>want to have to learn how to think mathematically, just give me ways I
>can subordinate mathematics to naive logic so that I can use it to
>extend my logical though whenever I feel like."
At 15, I did not have any concept that there was a difference between naive
logic and thinking mathematically. I'm not even sure I realized mathematics
was tied to being logical (naively or otherwise), until I learned
programming, wherein the stupidity of computers taught me that logic needs to
be broken down into individual steps that have to all be correct or the
program doesn't do what we want. I learned trivial programming in high
school, but didn't really learn what I just described until a college level
programming course, which was WAY too late to apply the lesson to the math
that I hadn't been properly learning for several years by then.
Now they try to get the idea that rigorous detail is important in programming
by teaching 4-6 graders to do Logo programming with "turtle graphics". I
don't know much about this, or even about Logo, and they don't spend a lot of
time on it any more here because it isn't in the state standards that are
tested.
>But that's an
>unrealistic proposition! Because mathematics isn't about expanding
>naive logic, it's about replacing it with mathematical thought. And
>the ability to think mathematically is not innate in many of us, and
>it takes years and years to develop it even to a minimum level.
Which means little to kids whose idea of a long term goal is what they will
do this weekend.
>>As I said, young kids have it, and so do older kids. My teenagers are just
>>as curious now as when they started school. But just as they were then,
>>their curiosity is shallow. They want a quick answer to their questions that
>>they do not have to consciously think about. Their minds are working
>>elsewhere, and the questions they ask are merely to fill a void so that the
>>world makes enough sense that they can think about whatever it is that is
>>really on their minds. My daughter constantly teases me because I cannot
>>give her an answer without a 5 minute (or more) explanation - what she wants
>>is a 5 WORD explanation, using no words or concepts that she doesn't already
>>know.
>
>The problem with math is this: quick answers come eventually, but not
>before we bend our intuition to be able to operate mathematically. If
>you need five minutes of explanation before you can reach for a result
>that would be intuitive otherwise, in math terms, maybe the problem is
>that your daughter lacks the minimum prerequisite to be able to use
>math in an effective way.
That is not just true for math with her, it is true with EVERYTHING
intellectual with her. She is curious, but not intellectually curious
(she'll listen for hours to people talking about their interpersonal
relationships because she "gets" that intuitively without having to think in
depth.
My son asks "why" more than she does. But he seldom can sit still for the
answer. And it is more or less true with every one of her friends that I've
talked with. We simply do not know how to change the childlike simple
curiosity into the deeper intellectual curiosity that is needed to plow
deeply into a subject. Some people develop natural intellectual curiosity,
usually through reading widely, others become very interested in one subject
because of an inspiring teacher or other adult, and then have the deeper
curiosity about that one subject. But most kids seem happier to just be
kids, and to get them into thinking deeply, you need to have the knack of
noticing the "teachable moment" when their interest level is piqued because
of some synergy of events.
>It reminds me the relationship between my wife and her computer.
...
Your wife is exactly about both my kids. They can't even keep the concepts
of Windows files and folders straight (neither of my kids is organized enough
to use folders usefully to keep things organized outside the computer, so the
metaphor is lost to them).
lojbab
Herman Rubin
>>It is studiable, but do the kids learn it. SMSG came about because the
>>Euclid based books that came before were NOT being understood.
>Just for the sake of my curiosity, what's SMSG ?
>Alberto.
School Mathematics Study Group. This was set up in the
post-Sputnik period to improve education, along with
other measures. Few of the measures suggested by the
committees of scholars whom the government asked for
suggestions were implemented. One which was consisted
of negating the educationists' policy of keeping the
parents uninformed, especially in the early grades; they
did not want parents contradicting the "priests".
Bob LeChevalier
>In article <707ciu084r3n7mhd2...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>>And, which few students seem to do,
>>>read the section to be discussed BEFORE coming to class.
>
>>That'd be nice, if we had understood the section before.
>
>School should be a place to learn. If you do not understand,
>ask questions! This should be encouraged from the beginning,
>not put off. The only stupid question is the one which is
>not asked.
The kid who asks lots of questions is put down NOT by the teacher, but by the
other kids (and mixing age groups won't much change this, because this is
largely independent of age). It is "uncool" to have any interest in getting
the attention of a teacher or other adult.
>>>>I didn't take college algebra. They discussed induction in high school
>>>>algebra, but I had no interest in proof, and never figured out why I should
>>>>care that they used induction to prove this or that.
>
>>>College algebra used to be standard in high school. It
>>>usually was part of the second year of algebra.
>
>>In other words, what we called Algebra II.
>
>In the old day, courses were by semesters, not years.
So, now they call it "Algebra II and Trigonometry" and it is a full year
course.
> I was the kid who they allowed to
>>do precisely what you call for - move ahead at whatever pace we could. I did;
>>had no problem doing any of the problems in the textbook including the proofs
>>except for the ones dealing with limits and infinite series, which I could
>>not understand at all (and never did).
>
>Limits are NEEDED to understand the "infinite decimals"
>usually taught around 6th grade.
Yes and no. "Rounding" solves the problem for the nonce.
>Without limits, there
>is no way to understand infinite series. You were
>supposedly there to learn; when you did not understand,
>and you knew it, you did nothing about it.
Correct. I worked with what I did understand. There was plenty to learn
expanding upon what I did understand, so that the stuff I didn't understand
just wasn't important enough to bother with.
>>>So you did not want to know why,
>
>>No. I did not CARE about why. For that matter I did not care about anything
>>other than whatever was explicitly stated in the textbook, because I was
>>adept at learning and grasping what was explicitly stated.
>
>You stated that you did not grasp what was explicitly stated,
>or you could have done those problems.
It was not explicitly stated in English, and I've never been any good at
learning foreign languages, math jargon included.
> You make the
>>assumption that most kids are curious to learn. In a sense they are, but
>>they usually are only curious about things that can be explained in 10
>>seconds or less: they want answers, data, not the need to think about
>>answers.
>
>And nothing important can be taught that way.
So what is the solution?
>It is not
>the answers which are important; it is asking the questions,
>and then figuring out how to get from there to the answer.
Convince the kids of that, and you might get somewhere.
>So you were a casualty of concentrating on memorization and
>routine, rather than understanding, which very often makes
>it unnecessary to memorize.
I memorized a lot, I understood a lot more, perhaps at what you would call
the "routine" level. Other kids had trouble even with the routine level.
> Duh, one is heavy and hits things,
>>and the other is flat with sharp teeth and cuts --- end of story (and yet now
>>I know that there ARE other properties of hammers and saws, which is why they
>>come in multiple shapes and sizes - but at age 15 all hammers are the same).
>
>That is why you need to get the understanding BEFORE age
>15. Once you get in the rut, it is hard to get out.
You've got the cart before the horse. You have to have questions in order to
seek deeper understanding. I did not have that sort of curiosity, and
neither did any other kid I knew. And it wasn't because parents and teachers
weren't trying to stimulate it in us.
>>>It was presented; that was enough reason to consider not
>>>just learning it, but understanding it.
>
>>The concept of learning anything more than what was explicitly stated in the
>>book was as of yet unconsidered by me.
>
>But it was explicitly stated in the book.
I learned what was explicitly stated. The "concepts" as you often note,
cannot be explicitly stated. They are implicit in the notation, not
explicit.
> It was my college English and history
>>classes that first suggested the possibility that my mind could come up with
>>useful and meaningful insights above and beyond what was explicitly told to
>>me in the book. Science and math classes gave the impression that we were
>>following the trail of mathematicians and scientists who had done it all
>>before, and the process of becoming like them was to do precisely what the
>>book told us to do. "Why?" was simply not a question one asked.
>
>The why was there; you just ignored it. You wanted
>simple answers, and fooled yourself into thinking that
>this was all that mattered.
Yep.
>>>Curiosity may have killed the cat, but not having it is
>>>deadly for education. Young children have it, but it seems
>>>to disappear. Could the way the schools approach things
>>>have anything to do with this?
>
>>As I said, young kids have it, and so do older kids. My teenagers are just
>>as curious now as when they started school. But just as they were then,
>>their curiosity is shallow. They want a quick answer to their questions that
>>they do not have to consciously think about. Their minds are working
>>elsewhere, and the questions they ask are merely to fill a void so that the
>>world makes enough sense that they can think about whatever it is that is
>>really on their minds. My daughter constantly teases me because I cannot
>>give her an answer without a 5 minute (or more) explanation - what she wants
>>is a 5 WORD explanation, using no words or concepts that she doesn't already
>>know.
>
>So this has to start early, not late, or the person's mind
>will be turned off from the real explanation.
You have to turn their mind ON. Kids have short attention spans, and their
minds operate on autopilot most of the time.
>>>Learning is for the FUTURE, not the present. Tying it to what
>>>is of present interest does great harm.
>
>>Kids live in the NOW.
>
>What is the purpose of education?
A pregnant question, that deserves more answer than I can give in the context
of this thread.
>The teachers who teach
>only for the NOW are at least weakening, if not destroying,
>their minds.
If the kids are only interested in "NOW", do the teachers have much choice.
> What are YOU, as a parent, doing about it?
Wishing I knew how to do something that no one else seems to know how to do:
get kids interested in the longer term.
>>>>I first saw that (algebra is not just for numbers) sometime around 1985, 11
>>>>years after finishing college.
>
>>>That is why it belongs in first grade, and not just for
>>>numbers.
>
>>I wouldn't have understood it then, and I couldn't explain it to my teenage
>>kids now in a way that would make sense to them. It takes experience before
>>you can get the idea that you can add things other than ordinary numbers and
>>that it will still be meaningful.
>
>Variables are a simple linguistic tool. They make it
>possible to communicate clearly about almost anything.
So you say. But kids already think that they communicate clearly. It's us
adults who make things complicated and unclear (and who realize that what
kids think is "clear" is nothing of the kind).
lojbab
Bob LeChevalier
>In article <lpgciuolkben7di0e...@4ax.com>,
>toto <scar...@wicked.witch> wrote:
>This is why mathematical and other precise subjects need to
>be taught early. Reading fiction does not teach someone
>how to learn.
>
>But they can read the texts for spelling, arithmetic, and
>geography even in the early grades. This makes reading a
>tool for learning subject matter, not just literature.
They read those texts in the same way that they read fiction: loosely. I've
tried asking my kids questions after they've read non-fiction, and it seems
like its just a story about the real world to them.
lojbab
Herman Rubin
toto <scar...@wicked.witch> wrote:
>>=====================
>>"You cannot expect to read mathematics the way you read a novel. If
>>you zip through a page in less than an hour, you are probably going
>>too fast. When you encounter the phrase "as you should verify", you
>>should indeed do the verification, which will usually require some
>>writing on your part. When steps are left out, you need to supply the
>>missing pieces. You should ponder and internalize each definition. For
>>each theorem, you should seek examples to show why each hypothesis is
>>necessary."
>>=====================
>And this is one of my points. We must begin to teach students how to
>read mathematics texts early on. We don't. We expect them to be
>able to figure out the techniques themselves and *most* kids can't
>do that.
I believe that one can handle mathematical texts faster
than stated more often than the author admits. As I
recall, the text was on linear algebra, which is usually
taken by students who have just undergone a cookbook
calculus course.
It is necessary to understand proofs and verify what is
left out, but many leave little out. In particular,
Landau's _Foundations of Analysis_ was considered to be
an easy read in the original German for those who knew
little German, until one gets to the construction of the
real numbers from the rationals.
Anyone who cannot read the part on the positive integers
easily should not be teaching first grade mathematics.
Bob LeChevalier
>>Euclid based books that came before were NOT being understood.
>
>
>Alberto.
School Mathematics Study Group. This was the group set up and funded by NSF
after Sputnik to try to fix mathematics education, and were the linchpin of
"New Math". They started around 1958 redesigning the high school textbooks,
then worked on the lower grades' textbooks, planning to then revise the upper
grade ones for the kids who used those textbooks. It seems that most of what
Herman has been saying took place in the 40s took place in the 60s, since
that is where the documents led from the clues that he has provided
mentioning Suppes and Stanford (which is where the SMSG was centered after
1961).
By the late 60s, parents were in open revolt because, if the kids might have
been learning mathematical concepts, they weren't learning to calculate, and
the parents did not understand the kids' homework. As Herman has said,
teachers did not understand it well, but the emphasis was only on teaching
the secondary teachers, and they did it in 6-8 week summer sessions that may
have taught concepts but did not convey a useful teaching methodology.
By 1971, SMSG was dead, having been rejected massively by local school
boards. At the same time a book came out called something like "Why Johnny
Can't Add" or something like that, paralleling the earlier "Why Johnny Can't
Read" that led to the "back-to-basics" phonics movement.
I gave a long post a couple days ago with quotes and cites about the history
of the New Math movement. If you read it, SMSG was the group that Yale
Professor Begle headed.
I'm kind of curious why the SMSG books are not available on line. According
to the histories, they were published in the public domain (which may have
hurt their acceptance, since textbook publishers couldn't make as much money
off them).
lojbab
Bob LeChevalier
>In article <7t9ciu4635ab310f9...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>In article <2d8aiu47hdib3tvb2...@4ax.com>,
>>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>>>The old Euclid books were used in the standard college
>>>>>preparatory program before. To some extent, they were
>>>>>even teacher proof, as theorems and proofs were carefully
>>>>>explained, but naturally there was nothing teaching
>>>>>students how to go about proving theorems.
>
>>>>Which was the problem with the SMSG book as well.
>
>>>Why is this a problem?
>
>>The kids come away having learned NOTHING.
>
>No, they came away with an understanding of proof, and
>geometry based on it.
I studied from the book, and NO, I did NOT come away with such an
understanding. I "knew" that geometry was about proofs, rather than about
geometric objects, and I knew what a proof was, but I had no idea why anyone
would care. And the fact that it was so unlike any prior math I had ever
studied meant that it had no connection to any other sort of mathematics in
my mind.
>>>For the high school material, there was not that much that
>>>the teachers were not supposed to have already had. The
>>>mathematics in the SMSG books was mainly already required,
>>>in many places even for college admission.
>
>>That which someone had several years ago in a course, and which has never
>>been used since, is usually GONE, and needs to be totally relearned.
>
>Some of the details might, but if the ideas need to be
>relearned, the student did not learn the course.
Most kids did not learn them in the first place. Few could do the proofs in
the SMSG problems; they didn't understand, and they usually didn't care to
understand. If they hadn't graded on a curve, I suspect that 75% of the
class would have flunked.
>>>Nobody expected that. They were to get the concepts, not
>>>the results of working with them.
>
>>You've said that there are math majors who manage to go through an entire
>>degree program and don't get the conceptual understanding they need, and
>>you've also said that you can't teach concepts. So how could you accomplish
>>it in 9 weeks or less?
>
>What I said is that they almost do not see the concepts.
>Did you understand limits after the calculus course?
I understood that limits were useful after calculus. I couldn't figure out
the limit of anything.
lojbab
toto
Rubin) wrote:
Herman, can you please listen - I am talking about k-12 here
and no elementary kid (k-3 say) is going to read Foundations
of Analysis. Are you going to tell me your son could have
handled this at 5 to 8? I don't think so
Dorothy
Bob LeChevalier
>On Sat, 06 Jul 2002 20:55:52 GMT, Bob LeChevalier
><loj...@lojban.org> wrote:
>
>>>This
>>>belongs well before college, and even before high school.
>>>Teaching why should be the rule, not the exception.
>>
>>They tried to teach us why. A derivation does not teach us why.
>
>I have to disagree with you here, Bob. In fact, the derivation
>goes to the *why* on a deep level. If you understand the why,
>then you can derive the formula when you need it. If you do
>not understand, seeing the derivation gives you the chance
>to get at the underlying why the formula works.
You missed my joke. A derivation gives one kind of why, but it is not the
"why" that any kid wants to know. "Why" is an ambiguous word in English
having a multitude of meanings (Bill Cosby observed that a PE teachers answer
to a kids "Why is there air?" was "To blow up volleyballs and basketballs".)
In Alberto's terms, the teachers satisfied our curiosity as to "why" with the
"naive reasoning" sort of explanation, and few kids bothered to try to figure
out the derivation, since they already knew "why". The derivation cannot
teach kids what they already think that they know.
lojbab
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <uq3ciuc2guui2u5qc...@4ax.com>,
>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
................
>Given that is has been YOU who's been touting Suppes' success in teaching
>younger kids math concepts that the teachers supposedly couldn't learn (I've
>found no evidence that he or anyone else tried to teach teachers using the
>same CAI methods, BTW), I would have presumed that you had some idea what it
>was that had been taught and how they taught it.
I am not familiar with his CAI methods. What I am familiar
with is what he did while he was actively teaching and doing
research, and I have pointed out weakneesses.
The paper I refer to below is 40 years old, and the results
are quite clear.
>>>Now you are saying that he in fact could not have been teaching them, and
>>>that they would have almost needed to start over.
>>If the concepts are presented first, and are required
>>to be used, excessive drill is only wasteful and boring.
>>Suppes is one of the authors of a paper which shows that
>>mathematical concept formation requires no time spent on
>>exercises after it is learned, and no benefit is gained.
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On Sat, 06 Jul 2002 20:55:52 GMT, Bob LeChevalier
><loj...@lojban.org> wrote:
>>>This
>>>belongs well before college, and even before high school.
>>>Teaching why should be the rule, not the exception.
>>They tried to teach us why. A derivation does not teach us why.
>I have to disagree with you here, Bob. In fact, the derivation
>goes to the *why* on a deep level. If you understand the why,
>then you can derive the formula when you need it. If you do
>not understand, seeing the derivation gives you the chance
>to get at the underlying why the formula works.
I have to partly agree with Bob here. There are many ways
to prove theorems. and some of them disguise understanding
almost completely. I have commented on teaching the
Neyman-Pearson Lemma, with the textbook proofs being quite
adequate as proofs, but providing no insight. They are
like using a computer program to compute the tangent of a
45 degree angle and say that it is 1, while just looking at
the problem will make that clear.
That is why we have to use concepts, preferably in addition
to proofs, to understand why.
Herman Rubin
toto <scar...@wicked.witch> wrote:
>On Sat, 06 Jul 2002 20:55:52 GMT, Bob LeChevalier
><loj...@lojban.org> wrote:
>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>In article <uq3ciuc2guui2u5qc...@4ax.com>,
>>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>>>>>A concept is not a set of words, to be memorized and
>>>>>>>regurgitated when requested. It is not a set of facts.
>>>>>>>It is not composed of procedures.
>>>>>>Irrelevant.
>>>>>NOT irrelevant. Those drilled in facts and procedures
>>>>>almost need to start over.
>>>>You seem to have a fundamental disagreement with Suppes, since apparently
>>>>that was precisely what he was using in his CAI courses used for teaching
>>>>bright kids, which you constantly hold up to show that it can be done. Any
>>>>course that has kids doing 100-150 problems in a class in one day is
>>>>precisely doing drill.
I have never stated that some practice in using concepts is
not essential. You will find that gifted people do more
than is needed in most cases, and many of them need very
little, as the "practice" can be a mental incorporation.
For how many word problems is it necessary to WRITE OUT the
formulation to understand the process of translation? It
greatly depends on the individual, and can vary from zero
to hundreds. Give a few hard ones; if they can be done,
stop drilling and just use it, but if not, shift to easier
ones, and build up.
What I said was that large amounts of drill in routine is
not necessary and does no good. The experiments conducted
by Suppes show that drill after a concept is learned do not
add to the understanding of the concept. The gifted
children who refused to learn the multiplication tables
have no less understanding of multiplication than if they
had learned them; memorizing the tables is at best a way
of possibly speeding up the process of getting answers.
Bob LeChevalier
>>School Mathematics Study Group. This was the group set up and funded by NSF
>>after Sputnik to try to fix mathematics education, and were the linchpin of
>>"New Math". They started around 1958...
>
>[snip...]
>
>Thanks, Bob, for the information. I have a much more pedestrian view
>of this kind of thing, so, I'm not sure I agree with Herman on his
>modern math posture. However, I do see the point, one of the battles I
>routinely fight is that of having to get my students to abandon the
>cozyness of traditional college math and think modern math instead.
>But I don't have solutions here, just questions.
The thing is, that it looks like, from what I read, that a lot of the New
Math ideas are, after a few decades of testing and developing ways to teach
them, the basis of the NCTM standards that you and Herman often criticize.
They are trying to move in the right direction, within the limits of what
they know how to teach to kids.
lojbab
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>Alberto Moreira <junk...@moreira.mv.com> wrote:
>>Bob LeChevalier <loj...@lojban.org> said:
>>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>>So learn from the textbook, and ask questions.
>>>I couldn't understand one paragraph of that textbook. One look at
>>>mathematical notation and my eyes glazed over. It made no sense at all to
>>>me.
>>Maybe you didn't have the prerequisite necessary to be able to read
>>the book ? Or maybe you didn't have the adequate level of skill to be
>>able to read mathematical notation in a more efficient way ? Or maybe
>>you lacked the willpower to put in the time to decode the mathematical
>>notation ?
>All of the above. No one in all my years of schooling EVER taught me to read
>mathematical notation efficiently, or even to read it at all. We were taught
>what the symbols meant,
There are a few constant symbols, like +, -, =, the digits, etc.
As for variables, they can be used for ANYTHING; they are "place
markers". It is the case that they cannot change meaning within
one context, but trying to make more out of this, which someone
who has been kept from it, or only sees mnemonic variables used,
will have difficulty. So, instead of having A = l x w, use
q = r x b. Mnemonic variables can speed things up, but should
not be used in the beginning for this reason.
and (sudden insight!) the situation with math
>notation for me and most students is precisely the same as a kid who learns
>via phonics to decode the symbols on the page and assemble them into words,
>but has no idea what many of the words mean, much less what entire phrases
>and sentences mean.
Do not expect to read it as quickly as literature. The amount
of content on one line is very much greater. It is this concise
syntax which enabled the advancement of mathematics. Euclid
used this for points and lines in his diagrams, and Aristotle
used it for propositions, but the use for numbers was later,
and the systematic use of more than one variable began in the
16th century.
As for what the words mean, in "q = r x b", the words "=" and
"x" have the meaning of equality and product, and the variables
have whatever meaning has been assigned to them in the previous
context. Initially, translate word for word into your language,
and then break out of this.
We never learn to read math better than the typical 1st
>grader reads English prose. It is presumed that once we know the symbols, we
>can figure out the rest.
Except for defined abbreviations, that is correct. A mathematical
statement says something concise, and the redundancy in English,
which enables scanning and quicker reading, is not present. There
is no way a spelling checker can be used with mathematical notation.
Other than using the rules of mathematical grammar, and the added
ones for the current context, there is no way to say that something
does not "make sense".
>>I believe I already passed this on, but I will repeat. It's from the
>>preface of Sheldon Axler's "Linear Algebra Done Right" book:
>>=====================
>>"You cannot expect to read mathematics the way you read a novel. If
>>you zip through a page in less than an hour, you are probably going
>>too fast. When you encounter the phrase "as you should verify", you
>>should indeed do the verification, which will usually require some
>>writing on your part. When steps are left out, you need to supply the
>>missing pieces. You should ponder and internalize each definition. For
>>each theorem, you should seek examples to show why each hypothesis is
>>necessary."
>>=====================
>All well and good that this is what "should be", but K12 students are NOT
>going to spend an hour reading a page, and will not verify anything (or do
>anything else) that they are not specifically assigned to verify by the
>teacher. A K12 teacher who expects the kids to spend more than an hour a day
>on only ONE subject will probably not keep his job. High school teachers
>today would probably be happy if kids consistently spent an hour a day total
>outside of class on their homework for ALL their classes, with the ideal high
>school kid perhaps spending 2 hours per day on homework.
We need to cut "class time" by half at least, and have
children study. Then, they might learn instead of just
memorize and regurgitate.
>>>No. I did not CARE about why. For that matter I did not care about anything>>>other than whatever was explicitly stated in the textbook, because I was
>>>adept at learning and grasping what was explicitly stated.
If you grasped the concepts, you would not be saying this. You
learned the words, and could repeat them. You learned the methods,
and could plug into them. If you did not know WHY, you cannot
state that you learned the material.
You make the
>>>assumption that most kids are curious to learn. In a sense they are, but
>>>they usually are only curious about things that can be explained in 10
>>>seconds or less: they want answers, data, not the need to think about
>>>answers.
>>If you don't care for proof, learning becomes parrotlike: you take
>>statements at face value, and you just repeat them. Proofs set up the
>>background for theorems, they give a justification for the result, and
>>they often develop additional material on the side that is important
>>to be digested if the student wants to understand what the theorem is
>>really all about. It is often the case that the proof conveys more
>>information than the theorem itself. So, if all you want is to read
>>something and grasp it within ten seconds, mathematics is indeed the
>>wrong field to study.
>I may understand this now, but at age 16 when I had had a history of grasping
>the math I had been taught in 10 seconds rather than an hour a page, it was
>not something I was prepared to accept even if someone had bothered to tell
>me.
I disagree with the hour a page; you could probably have
done it faster. For me, it was often a few minutes. You
might want to start with Landau, where you could almost
read it like prose.
>>>>and you cut yourself off from understanding the basic properties of the integers.
>>>Yes. And if you had put it to me like that, I would have wondered "why the
>>>heck should I care about the basic properties of the integers." The only
>>>purpose of numbers I understood is to get answers to questions of 'how many',
>>>and the only property I cared about was that they were useful tools for
>>>figuring out 'how many'.
>>But that requires knowing the basic properties of the integers. One
>>reason we study math is to replace naive reasoning with mathematical
>>reasoning.
>But of course no one ever said that. The purpose of studying math was to
>solve problems that used math. That there was a distinction between naive
>reasoning and mathematical reasoning never was stated. We were taught that
>there was this thing called "formal proof" where you write out all the steps,
>but that was something one did only when proving theorems, which nobody cared
>about anyway.
Here is a place where induction would show its value. You could
find the formula for the sum of the first n squares or cubes, and
quite a few other such things. Now of course you could memorize
those formulas, but if you forgot one, where would you be? Also,
you have the basis for finding others.
Use the binomial theorem, proved by induction, to show that
the more often interest is compounded, the greater the result.
There is no shortage of such problems.
That is because the elementary school teachers could not
understand that it is more. Even most high school teachers
have major problems with this.
>We have not learned how to teach kids that naive reasoning is not "good
>enough". And by the time they have learned it, the math is usually way over
>their head because they had run through years of classes being content with
>naive reasoning so long as it got the right answer, which it usually does at
>the lower levels.
Some of us have. However, those running the schools object
to introducing precise reasoning for two reasons. One is
that they do not understand it themselves; most people
never even encounter it except in computers, and most do
not understand that computers are just superfast imbeciles,
which do exactly what they are told, no matter how stupid.
The other is even worse. Politicians, social reformers,
and others like that realize that if the public was in the
habit of thinking logically, their ravings would be seen
for what they are, Redistribution of wealth will not have
everyone living as well as Bill Gates is living now. The
Universal Declaration of Rights passed a half century ago
is hopeless political propaganda, impossible to carry out.
Herman Rubin
Start the approach before they have this attitude. One can
test them in class on this, even from the beginning, just
as one tests on addition. There is some concern about their
grades, and even the parents who objected to them learning
mathematics which they did not know believe that they should
learn subject matter and not just sloppily.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>Alberto Moreira <junk...@moreira.mv.com> wrote:
>>Bob LeChevalier <loj...@lojban.org> said:
>>>It is studiable, but do the kids learn it. SMSG came about because the
>>>Euclid based books that came before were NOT being understood.
>>Just for the sake of my curiosity, what's SMSG ?
>>Alberto.
>School Mathematics Study Group. This was the group set up and funded by NSF
>after Sputnik to try to fix mathematics education, and were the linchpin of
>"New Math".
This was NOT as you say. The "new math" was started by a
professor whose daughter asked him a question, and he found
that she had no idea of what the arithmetic she could do well
meant. This was in the 40s.
The new math was started with elementary school material.
They started around 1958 redesigning the high school textbooks,
>then worked on the lower grades' textbooks, planning to then revise the upper
>grade ones for the kids who used those textbooks. It seems that most of what
>Herman has been saying took place in the 40s took place in the 60s, since
>that is where the documents led from the clues that he has provided
>mentioning Suppes and Stanford (which is where the SMSG was centered after
>1961).
As I said, SMSG was quite different from the new math movement.
Do not confuse the two. About the only thing in common was the
use of set algebra, which I do not believe is the best way to
approach the integers. It is a "cute" way.
>By the late 60s, parents were in open revolt because, if the kids might have
>been learning mathematical concepts, they weren't learning to calculate, and
>the parents did not understand the kids' homework.
This is true. What those who set up the new math did not realize
is that even the best understanding of concepts needs development
to do calculation at anything faster than the pace of a dying snail.
As Herman has said,
>teachers did not understand it well, but the emphasis was only on teaching
>the secondary teachers, and they did it in 6-8 week summer sessions that may
>have taught concepts but did not convey a useful teaching methodology.
Again, the new math was in the elementary schools. I have a
set of the version Suppes put out in the early 60s, starting
from the very beginning.
I have checked under the keywords "new math" in our library
catalog, and there were several entries from 1964. It will
be at least Tuesday before I can check any of them out.
>By 1971, SMSG was dead, having been rejected massively by local school
>boards. At the same time a book came out called something like "Why Johnny
>Can't Add" or something like that, paralleling the earlier "Why Johnny Can't
>Read" that led to the "back-to-basics" phonics movement.
"Why Johnny Can't Add" was written by Morris Kline, who
worked in classical analysis, and had no sympathy for
teaching axiomatic mathematics and set theory (NOT the set
algebra used in the books for the new math and SMSG) even
for the researchers in those fields.
>I gave a long post a couple days ago with quotes and cites about the history
>of the New Math movement. If you read it, SMSG was the group that Yale
>Professor Begle headed.
>I'm kind of curious why the SMSG books are not available on line. According
>to the histories, they were published in the public domain (which may have
>hurt their acceptance, since textbook publishers couldn't make as much money
>off them).
To put them on line, someone has to do the work. Nothing
was done using computer files in those days, so one has to
go back to typed, mimeographed, or printed materials and
do the work to converting them for the computer.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <7t9ciu4635ab310f9...@4ax.com>,
>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>>>In article <2d8aiu47hdib3tvb2...@4ax.com>,
>>>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>>>>The old Euclid books were used in the standard college
>>>>>>preparatory program before. To some extent, they were
>>>>>>even teacher proof, as theorems and proofs were carefully
>>>>>>explained, but naturally there was nothing teaching
>>>>>>students how to go about proving theorems.
>>>>>Which was the problem with the SMSG book as well.
>>>>Why is this a problem?
>>>The kids come away having learned NOTHING.
>>No, they came away with an understanding of proof, and
>>geometry based on it.
>I studied from the book, and NO, I did NOT come away with such an
>understanding. I "knew" that geometry was about proofs, rather than about
>geometric objects, and I knew what a proof was, but I had no idea why anyone
>would care. And the fact that it was so unlike any prior math I had ever
>studied meant that it had no connection to any other sort of mathematics in
>my mind.
I was exactly in the same situation; I had in the previous
summer seen an algebra book, learned it in a few days, and
it took about a month to get the instructor to test me out
of a full year of high school algebra. Then, moving to
another school (parents acquiring a mom and pop business),
the question came up what to do next. I had no idea of
what geometry was, and came into the class a month late.
It was totally unlike anything I had seen before, except
for "points", "lines", "triangles", and such. But it was
something to be learned and understood, just as any other
subject. I believed in learning.